2012
2012
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Paper 1, Section I, E
2012 commentWhat does it mean to say that a function is continuous at ?
Give an example of a continuous function which is bounded but attains neither its upper bound nor its lower bound.
The function is continuous and non-negative, and satisfies as and as . Show that is bounded above and attains its upper bound.
[Standard results about continuous functions on closed bounded intervals may be used without proof if clearly stated.]
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Paper 1, Section I,
2012 commentLet be continuous functions with for . Show that
where .
Prove there exists such that
[Standard results about continuous functions and their integrals may be used without proof, if clearly stated.]
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Paper 1, Section II, E
2012 comment(a) What does it mean to say that the sequence of real numbers converges to
Suppose that are sequences of real numbers converging to the same limit . Let be a sequence such that for every ,
Show that also converges to .
Find a collection of sequences such that for every but the sequence defined by diverges.
(b) Let be real numbers with . Sequences are defined by and
Show that and converge to the same limit.
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Paper 1, Section II, D
2012 commentLet be a sequence of reals.
(i) Show that if the sequence is convergent then so is the sequence .
(ii) Give an example to show the sequence being convergent does not imply that the sequence is convergent.
(iii) If as for each positive integer , does it follow that is convergent? Justify your answer.
(iv) If as for every function from the positive integers to the positive integers, does it follow that is convergent? Justify your answer.
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Paper 1, Section II, D
2012 commentLet be a continuous function from to such that for every . We write for the -fold composition of with itself (so for example .
(i) Prove that for every we have as .
(ii) Must it be the case that for every there exists with the property that for all ? Justify your answer.
Now suppose that we remove the condition that be continuous.
(iii) Give an example to show that it need not be the case that for every we have as .
(iv) Must it be the case that for some we have as ? Justify your answer.
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Paper 1, Section II, F
2012 comment(a) (i) State the ratio test for the convergence of a real series with positive terms.
(ii) Define the radius of convergence of a real power series .
(iii) Prove that the real power series and have equal radii of convergence.
(iv) State the relationship between and within their interval of convergence.
(b) (i) Prove that the real series
have radius of convergence .
(ii) Show that they are differentiable on the real line , with and , and deduce that .
[You may use, without proof, general theorems about differentiating within the interval of convergence, provided that you give a clear statement of any such theorem.]
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Paper 2, Section I, A
2012 commentFind two linearly independent solutions of
Find the solution in of
subject to at .
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Paper 2, Section I, A
2012 commentFind the constant solutions (those with ) of the discrete equation
and determine their stability.
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Paper 2, Section II, A
2012 commentFind the first three non-zero terms in the series solutions and for the differential equation
that satisfy
Identify these solutions in closed form.
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Paper 2, Section II, A
2012 commentConsider the function
Find the critical (stationary) points of . Determine the type of each critical point. Sketch the contours of constant.
Now consider the coupled differential equations
Show that is a non-increasing function of . If and at , where does the solution tend to as ?
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Paper 2, Section II, A
2012 commentFind the solution to the system of equations
in subject to
[Hint: powers of t.]
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Paper 2, Section II, A
2012 commentConsider the second-order differential equation for in
(i) For , find the general solution of .
(ii) For with , find the solution of that satisfies and at .
(iii) For with , find the solution of that satisfies and at
(iv) Show that
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Paper 4, Section I, B
2012 commentTwo particles of masses and have position vectors and respectively. Particle 2 exerts a force ) on particle 1 (where ) and there are no external forces.
Prove that the centre of mass of the two-particle system will move at constant speed along a straight line.
Explain how the two-body problem of determining the motion of the system may be reduced to that of a single particle moving under the force .
Suppose now that and that
is gravitational attraction. Let be a circle fixed in space. Is it possible (by suitable choice of initial conditions) for the two particles to be traversing at the same constant angular speed? Give a brief reason for your answer.
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Paper 4, Section I, B
2012 commentLet and be inertial frames in 2-dimensional spacetime with coordinate systems and respectively. Suppose that moves with positive velocity relative to and the spacetime origins of and coincide. Write down the Lorentz transformation relating the coordinates of any event relative to the two frames.
Show that events which occur simultaneously in are not generally seen to be simultaneous when viewed in .
In two light sources and are at rest and placed a distance apart. They simultaneously each emit a photon in the positive direction. Show that in the photons are separated by a constant distance .
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Paper 4, Section II, B
2012 commentLet be polar coordinates in the plane. A particle of mass moves in the plane under an attractive force of towards the origin . You may assume that the acceleration a is given by
where and are the unit vectors in the directions of increasing and respectively, and the dot denotes .
(a) Show that is a constant of the motion. Introducing show that and derive the geometric orbit equation
(b) Suppose now that
and that initially the particle is at distance from , moving with speed in a direction making angle with the radial vector pointing towards .
Show that and find as a function of . Hence or otherwise show that the particle returns to its original position after one revolution about and then flies off to infinity.
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Paper 4, Section II, B
2012 commentFor any frame and vector , let denote the derivative of relative to . A frame of reference rotates with constant angular velocity with respect to an inertial frame and the two frames have a common origin . [You may assume that for any vector
(a) If is the position vector of a point from , show that
where is the velocity in .
Suppose now that is the position vector of a particle of mass moving under a conservative force and a force that is always orthogonal to the velocity in . Show that the quantity
is a constant of the motion. [You may assume that .]
(b) A bead slides on a frictionless circular hoop of radius which is forced to rotate with constant angular speed about a vertical diameter. Let denote the angle between the line from the centre of the hoop to the bead and the downward vertical. Using the results of (a), or otherwise, show that
Deduce that if there are two equilibrium positions off the axis of rotation, and show that these are stable equilibria.
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Paper 4, Section II, B
2012 comment(a) State the parallel axis theorem for moments of inertia.
(b) A uniform circular disc of radius and total mass can turn frictionlessly about a fixed horizontal axis that passes through a point on its circumference and is perpendicular to its plane. Initially the disc hangs at rest (in constant gravity ) with its centre being vertically below . Suppose the disc is disturbed and executes free oscillations. Show that the period of small oscillations is .
(c) Suppose now that the disc is released from rest when the radius is vertical with directly above . Find the angular velocity and angular acceleration of about when the disc has turned through angle . Let denote the reaction force at on the disc. Find the acceleration of the centre of mass of the disc. Hence, or otherwise, show that the component of parallel to is .
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Paper 4, Section II, B
2012 comment(a) Define the 4-momentum of a particle of rest mass and 3 -velocity , and the 4-momentum of a photon of frequency (having zero rest mass) moving in the direction of the unit vector .
Show that if and are timelike future-pointing 4-vectors then (where the dot denotes the Lorentz-invariant scalar product). Hence or otherwise show that the law of conservation of 4 -momentum forbids a photon to spontaneously decay into an electron-positron pair. [Electrons and positrons have equal rest masses .]
(b) In the laboratory frame an electron travelling with velocity u collides with a positron at rest. They annihilate, producing two photons of frequencies and that move off at angles and to , in the directions of the unit vectors and respectively. By considering 4-momenta in the laboratory frame, or otherwise, show that
where
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Paper 3, Section I, E
2012 commentState Lagrange's Theorem. Deduce that if is a finite group of order , then the order of every element of is a divisor of .
Let be a group such that, for every . Show that is abelian. Give an example of a non-abelian group in which every element satisfies .
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Paper 3, Section I, E
2012 commentWhat is a cycle in the symmetric group ? Show that a cycle of length and a cycle of length in are conjugate if and only if .
Suppose that is odd. Show that any two -cycles in are conjugate. Are any two 3 -cycles in conjugate? Justify your answer.
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Paper 3, Section II, E
2012 comment(i) State and prove the Orbit-Stabilizer Theorem.
Show that if is a finite group of order , then is isomorphic to a subgroup of the symmetric group .
(ii) Let be a group acting on a set with a single orbit, and let be the stabilizer of some element of . Show that the homomorphism given by the action is injective if and only if the intersection of all the conjugates of equals .
(iii) Let denote the quaternion group of order 8 . Show that for every is not isomorphic to a subgroup of .
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Paper 3, Section II, E
2012 commentLet be , the groups of real matrices of determinant 1 , acting on by Möbius transformations.
For each of the points , compute its stabilizer and its orbit under the action of . Show that has exactly 3 orbits in all.
Compute the orbit of under the subgroup
Deduce that every element of may be expressed in the form where and for some ,
How many ways are there of writing in this form?
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Paper 3, Section II,
2012 commentLet be the set of (residue classes of) integers , and let
Show that is a group under multiplication. [You may assume throughout this question that multiplication of matrices is associative.]
Let be the set of 2-dimensional column vectors with entries in . Show that the mapping given by
is a group action.
Let be an element of order . Use the orbit-stabilizer theorem to show that there exist , not both zero, with
Deduce that is conjugate in to the matrix
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Paper 3, Section II, E
2012 commentLet be a prime number, and an integer with . Let be the Cartesian product
Show that the binary operation
where
makes into a group. Show that is abelian if and only if .
Let and be the subsets
of . Show that is a normal subgroup of , and that is a subgroup which is normal if and only if .
Find a homomorphism from to another group whose kernel is .
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Paper 4, Section I, D
2012 comment(i) Find integers and such that .
(ii) Find an integer such that and .
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Paper 4, Section I,
2012 commentWhat is an equivalence relation on a set ? If is an equivalence relation on , what is an equivalence class of ? Prove that the equivalence classes of form a partition of .
Let and be equivalence relations on a set . Which of the following are always equivalence relations? Give proofs or counterexamples as appropriate.
(i) The relation on given by if both and .
(ii) The relation on given by if or .
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Paper 4, Section II, D
2012 commentLet be a set, and let and be functions from to . Which of the following are always true and which can be false? Give proofs or counterexamples as appropriate.
(i) If is the identity map then is the identity map.
(ii) If then is the identity map.
(iii) If then is the identity map.
How (if at all) do your answers change if we are given that is finite?
Determine which sets have the following property: if is a function from to such that for every there exists a positive integer with , then there exists a positive integer such that is the identity map. [Here denotes the -fold composition of with itself.]
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Paper 4, Section II, D
2012 commentState Fermat's Theorem and Wilson's Theorem.
For which prime numbers does the equation have a solution? Justify your answer.
For a prime number , and an integer that is not a multiple of , the order of is the least positive integer such that . Show that if has order and also then must divide .
For a positive integer , let . If is a prime factor of , determine the order of . Hence show that the are pairwise coprime.
Show that if is a prime of the form then cannot be a factor of any . Give, with justification, a prime of the form such that is not a factor of any .
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Paper 4, Section II, D
2012 commentProve that each of the following numbers is irrational: (i) (ii) (iii) The real root of the equation (iv) .
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Paper 4, Section II, D
2012 commentShow that there is no injection from the power-set of to . Show also that there is an injection from to .
Let be the set of all functions from to such that for all but finitely many . Determine whether or not there exists an injection from to .
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Paper 2, Section I, F
2012 commentGiven two events and with and , define the conditional probability .
Show that
A random number of fair coins are tossed, and the total number of heads is denoted by . If for , find .
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Paper 2, Section I, F
2012 commentDefine the probability generating function of a random variable taking values in the non-negative integers.
A coin shows heads with probability on each toss. Let be the number of tosses up to and including the first appearance of heads, and let . Find the probability generating function of .
Show that where .
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Paper 2, Section II, F
2012 comment(i) Define the moment generating function of a random variable . If are independent and , show that the moment generating function of is .
(ii) Assume , and for . Explain the expansion
where and [You may assume the validity of interchanging expectation and differentiation.]
(iii) Let be independent, identically distributed random variables with mean 0 and variance 1 , and assume their moment generating function satisfies the condition of part (ii) with .
Suppose that and are independent. Show that , and deduce that satisfies .
Show that as , and deduce that for all .
Show that and are normally distributed.
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Paper 2, Section II, F
2012 comment(i) Define the distribution function of a random variable , and also its density function assuming is differentiable. Show that
(ii) Let be independent random variables each with the uniform distribution on . Show that
What is the probability that the random quadratic equation has real roots?
Given that the two roots of the above quadratic are real, what is the probability that both and
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Paper 2, Section II, F
2012 comment(i) Let be the size of the generation of a branching process with familysize probability generating function , and let . Show that the probability generating function of satisfies for .
(ii) Suppose the family-size mass function is Find , and show that
Deduce the value of .
(iii) Write down the moment generating function of . Hence or otherwise show that, for ,
[You may use the continuity theorem but, if so, should give a clear statement of it.]
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Paper 2, Section II, F
2012 commentLet be independent random variables with distribution functions . Show that have distribution functions
Now let be independent random variables, each having the exponential distribution with parameter 1. Show that has the exponential distribution with parameter 2 , and that is independent of .
Hence or otherwise show that has the same distribution as , and deduce the mean and variance of .
[You may use without proof that has mean 1 and variance 1.]
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Paper 3, Section I, C
2012 commentDefine what it means for a differential to be exact, and derive a necessary condition on and for this to hold. Show that one of the following two differentials is exact and the other is not:
Show that the differential which is not exact can be written in the form for functions and , to be determined.
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Paper 3, Section I, C
2012 commentWhat does it mean for a second-rank tensor to be isotropic? Show that is isotropic. By considering rotations through about the coordinate axes, or otherwise, show that the most general isotropic second-rank tensor in has the form , for some scalar .
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Paper 3, Section II, C
2012 commentState Stokes' Theorem for a vector field on .
Consider the surface defined by
Sketch the surface and calculate the area element in terms of suitable coordinates or parameters. For the vector field
compute and calculate .
Use Stokes' Theorem to express as an integral over and verify that this gives the same result.
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Paper 3, Section II, C
2012 commentConsider the transformation of variables
Show that the interior of the unit square in the plane
is mapped to the interior of the unit square in the plane,
[Hint: Consider the relation between and when , for constant.]
Show that
Now let
By calculating
as a function of and , or otherwise, show that
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Paper 3, Section II, C
2012 comment(a) Prove the identity
(b) If is an irrotational vector field (i.e. everywhere), prove that there exists a scalar potential such that .
Show that the vector field
is irrotational, and determine the corresponding potential .
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Paper 3, Section II, C
2012 comment(i) Let be a bounded region in with smooth boundary . Show that Poisson's equation in
has at most one solution satisfying on , where and are given functions.
Consider the alternative boundary condition on , for some given function , where is the outward pointing normal on . Derive a necessary condition in terms of and for a solution of Poisson's equation to exist. Is such a solution unique?
(ii) Find the most general spherically symmetric function satisfying
in the region for . Hence in each of the following cases find all possible solutions satisfying the given boundary condition at : (a) , (b) .
Compare these with your results in part (i).
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Paper 1, Section I,
2012 comment(a) Let be the set of all with real part 1 . Draw a picture of and the image of under the map in the complex plane.
(b) For each of the following equations, find all complex numbers which satisfy it:
(i) ,
(ii) .
(c) Prove that there is no complex number satisfying .
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Paper 1, Section I, A
2012 commentDefine what is meant by the terms rotation, reflection, dilation and shear. Give examples of real matrices representing each of these.
Consider the three matrices
Identify the three matrices in terms of your definitions above.
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Paper 1, Section II,
2012 commentThe equation of a plane in is
where is a constant scalar and is a unit vector normal to . What is the distance of the plane from the origin ?
A sphere with centre and radius satisfies the equation
Show that the intersection of and contains exactly one point if .
The tetrahedron is defined by the vectors , and with . What does the condition imply about the set of vectors ? A sphere with radius lies inside the tetrahedron and intersects each of the three faces , and in exactly one point. Show that the centre of satisfies
Given that the vector is orthogonal to the plane of the face , obtain an equation for . What is the distance of from the origin?
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Paper 1, Section II, A
2012 commentExplain why the number of solutions of the simultaneous linear equations is 0,1 or infinity, where is a real matrix and and are vectors in . State necessary and sufficient conditions on and for each of these possibilities to hold.
Let and be real matrices. Give necessary and sufficient conditions on for there to exist a unique real matrix satisfying .
Find when
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Paper 1, Section II, 7B
2012 comment(a) Consider the matrix
Determine whether or not is diagonalisable.
(b) Prove that if and are similar matrices then and have the same eigenvalues with the same corresponding algebraic multiplicities.
Is the converse true? Give either a proof (if true) or a counterexample with a brief reason (if false).
(c) State the Cayley-Hamilton theorem for a complex matrix and prove it in the case when is a diagonalisable matrix.
Suppose that an matrix has for some (where denotes the zero matrix). Show that .
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Paper 1, Section II, B
2012 comment(a) (i) Find the eigenvalues and eigenvectors of the matrix
(ii) Show that the quadric in defined by
is an ellipsoid. Find the matrix of a linear transformation of that will map onto the unit sphere .
(b) Let be a real orthogonal matrix. Prove that:
(i) as a mapping of vectors, preserves inner products;
(ii) if is an eigenvalue of then and is also an eigenvalue of .
Now let be a real orthogonal matrix having as an eigenvalue of algebraic multiplicity 2. Give a geometrical description of the action of on , giving a reason for your answer. [You may assume that orthogonal matrices are always diagonalisable.]
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Paper 3, Section I,
2012 commentLet be the set of continuous real-valued functions on with the uniform norm. Suppose is defined by
for all and . Is a contraction mapping? Does have a unique fixed point? Justify your answers.
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Paper 4, Section I, E
2012 commentLet be a bilinear function. Show that is differentiable at any point in and find its derivative.
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Paper 2, Section I, E
2012 commentLet be a function. What does it mean to say that is differentiable at a point Prove directly from this definition, that if is differentiable at , then is continuous at .
Let be the function:
For which points is differentiable? Justify your answer.
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Paper 1, Section II, E
2012 commentState the inverse function theorem for a function . Suppose is a differentiable bijection with also differentiable. Show that the derivative of at any point in is a linear isomorphism.
Let be a function such that the partial derivatives exist and are continuous. Assume there is a point for which and . Prove that there exist open sets and containing and , respectively, such that for every there exists a unique such that and . Moreover, if we define by , prove that is differentiable with continuous derivative. Find the derivative of at in terms of and .
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Paper 4, Section II, E
2012 commentState and prove the Bolzano-Weierstrass theorem in . [You may assume the Bolzano-Weierstrass theorem in .]
Let be a subset and let be a mapping such that for all , where is the Euclidean distance in . Prove that if is closed and bounded, then is a bijection. Is this result still true if we drop the boundedness assumption on ? Justify your answer.
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Paper 3, Section II, E
2012 commentLet be a sequence of continuous functions on the interval such that for each . For the three statements:
(a) uniformly on ;
(b) is a continuous function;
(c) as
say which of the six possible implications , are true and which false, giving in each case a proof or counterexample.
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Paper 2, Section II, E
2012 commentLet be a mapping. Fix and prove that the following two statements are equivalent:
(i) Given there is such that whenever (we use the standard norm in Euclidean space).
(ii) for any sequence .
We say that is continuous if (i) (or equivalently (ii)) holds for every .
Let and be subsets of and respectively. For as above, determine which of the following statements are always true and which may be false, giving a proof or a counterexample as appropriate.
(a) If is closed whenever is closed, then is continuous.
(b) If is continuous, then is closed whenever is closed.
(c) If is continuous, then is open whenever is open.
(d) If is continuous, then is bounded whenever is bounded.
(e) If is continuous and is bounded whenever is bounded, then is closed whenever is closed.
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Paper 4, Section I,
2012 commentLet be a holomorphic function with . Does there exist a holomorphic function defined in for which ? Does there exist a holomorphic function defined in for which ? Justify your answers.
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Paper 2, Section II, A
2012 commentBy a suitable choice of contour show that, for ,
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Paper 3, Section II, E
2012 commentLet denote the disc and let be a holomorphic function. Using Cauchy's integral formula show that for every
Deduce that if for every , then is constant.
Let be holomorphic with . Show that for all . Moreover, show that if for some , then there exists with such that for all .
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Paper 1, Section I, A
2012 commentFind a conformal transformation that maps the domain , on to the strip .
Hence find a bounded harmonic function on subject to the boundary conditions on , respectively, where is a real constant.
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Paper 1, Section II, 13A
2012 commentUsing Cauchy's integral theorem, write down the value of a holomorphic function where in terms of a contour integral around the unit circle,
By considering the point , or otherwise, show that
By setting , show that for any harmonic function ,
if .
Assuming that the function , which is the conjugate harmonic function to , can be written as
deduce that
[You may use the fact that on the unit circle, , and hence
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Paper 3, Section I, A
2012 commentState the formula for the Laplace transform of a function , defined for .
Let be periodic with period (i.e. ). If is defined to be equal to in and zero elsewhere and its Laplace transform is , show that the Laplace transform of is given by
Hence, or otherwise, find the inverse Laplace transform of
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Paper 4, Section II, A
2012 commentState the convolution theorem for Fourier transforms.
The function satisfies
on the half-plane , subject to the boundary conditions
Using Fourier transforms, show that
and hence that
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Paper 2, Section I, B
2012 commentWrite down the expressions for a general, time-dependent electric field and magnetic field in terms of a vector potential and scalar potential . What is meant by a gauge transformation of and ? Show that and are unchanged under a gauge transformation.
A plane electromagnetic wave has vector and scalar potentials
where and are constants. Show that can be modified to by a gauge transformation. What choice of leads to the modified satisfying the Coulomb gauge condition ?
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Paper 4, Section I, B
2012 commentDefine the notions of magnetic flux, electromotive force and resistance, in the context of a single closed loop of wire. Use the Maxwell equation
to derive Faraday's law of induction for the loop, assuming the loop is at rest.
Suppose now that the magnetic field is where is a constant, and that the loop of wire, with resistance , is a circle of radius a lying in the plane. Calculate the current in the wire as a function of time.
Explain briefly why, even in a time-independent magnetic field, an electromotive force may be produced in a loop of wire that moves through the field, and state the law of induction in this situation.
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Paper 1, Section II, B
2012 commentA sphere of radius a carries an electric charge uniformly distributed over its surface. Calculate the electric field outside and inside the sphere. Also calculate the electrostatic potential outside and inside the sphere, assuming it vanishes at infinity. State the integral formula for the energy of the electric field and use it to evaluate as a function of
Relate to the potential on the surface of the sphere and explain briefly the physical interpretation of the relation.
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Paper 3, Section II, B
2012 commentUsing the Maxwell equations
show that in vacuum, E satisfies the wave equation
where , as well as . Also show that at a planar boundary between two media, (the tangential component of ) is continuous. Deduce that if one medium is of negligible resistance, .
Consider an empty cubic box with walls of negligible resistance on the planes , , where . Show that an electric field in the interior of the form
with and positive integers, satisfies the boundary conditions on all six walls. Now suppose that
where and are constants. Show that the wave equation is satisfied, and determine the frequency . Find the further constraint on and ?
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Paper 2, Section II, B
2012 commentA straight wire has mobile, charged particles per unit length, each of charge . Assuming the charges all move with velocity along the wire, show that the current is .
Using the Lorentz force law, show that if such a current-carrying wire is placed in a uniform magnetic field of strength perpendicular to the wire, then the force on the wire, per unit length, is .
Consider two infinite parallel wires, with separation , carrying (in the same sense of direction) positive currents and , respectively. Find the force per unit length on each wire, determining both its magnitude and direction.
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Paper 1, Section I, A
2012 commentViscous fluid, with viscosity and density flows along a straight circular pipe of radius . The average velocity of the flow is . Define a Reynolds number for the flow.
The flow is driven by a constant pressure gradient along the pipe and the velocity is parallel to the axis of the pipe with magnitude that satisfies
where is the radial distance from the axis.
State the boundary conditions on and find the velocity as a function of assuming that it is finite on the axis . Hence, show that the shear stress at the pipe wall is independent of the viscosity. Why is this the case?
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Paper 2, Section I, A
2012 commentStarting from Euler's equation for the motion of an inviscid fluid, derive the vorticity equation in the form
Deduce that an initially irrotational flow remains irrotational.
Consider a plane flow that at time is described by the streamfunction
Calculate the vorticity everywhere at times .
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Paper 1, Section II, A
2012 commentConsider inviscid, incompressible fluid flow confined to the plane. The fluid has density , and gravity can be neglected. Using the conservation of volume flux, determine the velocity potential of a point source of strength , in terms of the distance from the source.
Two point sources each of strength are located at and . Find the velocity potential of the flow.
Show that the flow in the region is equivalent to the flow due to a source at and a fixed boundary at
Find the pressure on the boundary and hence determine the force on the boundary.
[Hint: you may find the substitution useful for the calculation of the pressure.]
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Paper 4, Section II, A
2012 commentThe equations governing the flow of a shallow layer of inviscid liquid of uniform depth rotating with angular velocity about the vertical -axis are
where are the - and -components of velocity, respectively, and is the elevation of the free surface. Show that these equations imply that
Consider an initial state where there is flow in the -direction given by
Find the initial potential vorticity.
Show that when this initial state adjusts, there is a final steady state independent of in which satisfies
where .
In the case , find the final free surface elevation that is finite at large and which is continuous and has continuous slope at , and show that it is negative for all .
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Paper 3, Section II, A
2012 commentA rigid circular cylinder of radius executes small amplitude oscillations with velocity in a direction perpendicular to its axis, while immersed in an inviscid fluid of density contained within a larger concentric fixed cylinder of radius . Gravity is negligible. Neglecting terms quadratic in the amplitude, determine the boundary condition on the velocity on the inner cylinder, and calculate the velocity potential of the induced flow.
With the same approximations show that the difference in pressures on the surfaces of the two cylinders has magnitude
where is the azimuthal angle measured from the direction of .
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Paper 1, Section I, G
2012 commentDescribe a collection of charts which cover a circular cylinder of radius . Compute the first fundamental form, and deduce that the cylinder is locally isometric to the plane.
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Paper 3, Section I,
2012 commentState a formula for the area of a hyperbolic triangle.
Hence, or otherwise, prove that if and are disjoint geodesics in the hyperbolic plane, there is at most one geodesic which is perpendicular to both and .
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Paper 2, Section II, G
2012 commentLet be a closed surface, equipped with a triangulation. Define the Euler characteristic of . How does depend on the triangulation?
Let and denote the number of vertices, edges and faces of the triangulation. Show that .
Suppose now the triangulation is tidy, meaning that it has the property that no two vertices are joined by more than one edge. Deduce that satisfies
Hence compute the minimal number of vertices of a tidy triangulation of the real projective plane. [Hint: it may be helpful to consider the icosahedron as a triangulation of the sphere
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Paper 3, Section II, G
2012 commentDefine the first and second fundamental forms of a smooth surface , and explain their geometrical significance.
Write down the geodesic equations for a smooth curve . Prove that is a geodesic if and only if the derivative of the tangent vector to is always orthogonal to .
A plane cuts in a smooth curve , in such a way that reflection in the plane is an isometry of (in particular, preserves ). Prove that is a geodesic.
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Paper 4, Section II, G
2012 commentLet be a smooth closed surface. Define the principal curvatures and at a point . Prove that the Gauss curvature at is the product of the two principal curvatures.
A point is called a parabolic point if at least one of the two principal curvatures vanishes. Suppose is a plane and is tangent to along a smooth closed curve . Show that is composed of parabolic points.
Can both principal curvatures vanish at a point of ? Briefly justify your answer.
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Paper 3, Section I,
2012 commentWhat is a Euclidean domain?
Giving careful statements of any general results you use, show that in the ring is irreducible but not prime.
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Paper 2, Section I,
2012 commentWhat does it mean to say that the finite group acts on the set ?
By considering an action of the symmetry group of a regular tetrahedron on a set of pairs of edges, show there is a surjective homomorphism .
[You may assume that the symmetric group is generated by transpositions.]
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Paper 4, Section I,
2012 commentAn idempotent element of a ring is an element satisfying . A nilpotent element is an element e satisfying for some .
Let be non-zero. In the ring , can the polynomial be (i) an idempotent, (ii) a nilpotent? Can satisfy the equation ? Justify your answers.
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Paper 1, Section II, G
2012 commentLet be a finite group. What is a Sylow -subgroup of ?
Assuming that a Sylow -subgroup exists, and that the number of conjugates of is congruent to , prove that all Sylow -subgroups are conjugate. If denotes the number of Sylow -subgroups, deduce that
If furthermore is simple prove that either or
Deduce that a group of order cannot be simple.
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Paper 2, Section II, G
2012 commentState Gauss's Lemma. State Eisenstein's irreducibility criterion.
(i) By considering a suitable substitution, show that the polynomial is irreducible over .
(ii) By working in , show that the polynomial is irreducible over .
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Paper 3, Section II, G
2012 commentFor each of the following assertions, provide either a proof or a counterexample as appropriate:
(i) The ring is a field.
(ii) The ring is a field.
(iii) If is a finite field, the ring contains irreducible polynomials of arbitrarily large degree.
(iv) If is the ring of continuous real-valued functions on the interval , and the non-zero elements satisfy and , then there is some unit with .
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Paper 4, Section II, G
2012 commentLet be a commutative ring with unit 1. Prove that an -module is finitely generated if and only if it is a quotient of a free module , for some .
Let be a finitely generated -module. Suppose now is an ideal of , and is an -homomorphism from to with the property that
Prove that satisfies an equation
where each . [You may assume that if is a matrix over , then (id), with id the identity matrix.]
Deduce that if satisfies , then there is some satisfying
Give an example of a finitely generated -module and a proper ideal of satisfying the hypothesis , and for your example, give an explicit such element .
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Paper 4, Section I, F
2012 commentLet be a complex vector space with basis . Define by for and . Show that is diagonalizable and find its eigenvalues. [You may use any theorems you wish, as long as you state them clearly.]
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Paper 2, Section I,
2012 commentDefine the determinant of an real matrix . Suppose that is a matrix with block form
where and are matrices of dimensions and respectively. Show that .
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Paper 1, Section I, F
2012 commentDefine the notions of basis and dimension of a vector space. Prove that two finitedimensional real vector spaces with the same dimension are isomorphic.
In each case below, determine whether the set is a basis of the real vector space
(i) is the complex numbers; .
(ii) is the vector space of all polynomials in with real coefficients;
(iii) , where
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Paper 1, Section II, F
2012 commentDefine what it means for two matrices to be similar to each other. Show that if two matrices are similar, then the linear transformations they define have isomorphic kernels and images.
If and are real matrices, we define . Let
Show that and are linear subspaces of . If and are similar, show that and .
Suppose that is diagonalizable and has characteristic polynomial
where . What are and
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Paper 4, Section II, F
2012 commentLet be a finite-dimensional real vector space of dimension . A bilinear form is nondegenerate if for all in , there is some with . For , define . Assuming is nondegenerate, show that whenever .
Suppose that is a nondegenerate, symmetric bilinear form on . Prove that there is a basis of with for . [If you use the fact that symmetric matrices are diagonalizable, you must prove it.]
Define the signature of a quadratic form. Explain how to determine the signature of the quadratic form associated to from the basis you constructed above.
A linear subspace is said to be isotropic if for all . Show that if is nondegenerate, the maximal dimension of an isotropic subspace of is , where is the signature of the quadratic form associated to .
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Paper 3, Section II, F
2012 commentWhat is meant by the Jordan normal form of an complex matrix?
Find the Jordan normal forms of the following matrices:
Suppose is an invertible complex matrix. Explain how to derive the characteristic and minimal polynomials of from the characteristic and minimal polynomials of . Justify your answer. [Hint: write each polynomial as a product of linear factors.]
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Paper 2, Section II, F
2012 comment(i) Define the transpose of a matrix. If and are finite-dimensional real vector spaces, define the dual of a linear map . How are these two notions related?
Now suppose and are finite-dimensional inner product spaces. Use the inner product on to define a linear map and show that it is an isomorphism. Define the adjoint of a linear map . How are the adjoint of and its dual related? If is a matrix representing , under what conditions is the adjoint of represented by the transpose of ?
(ii) Let be the vector space of continuous real-valued functions on , equipped with the inner product
Let be the linear map
What is the adjoint of
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Paper 3, Section I, H
2012 commentA runner owns pairs of running shoes and runs twice a day. In the morning she leaves her house by the front door, and in the evening she leaves by the back door. On starting each run she looks for shoes by the door through which she exits, and runs barefoot if none are there. At the end of each run she is equally likely to return through the front or back doors. She removes her shoes (if any) and places them by the door. In the morning of day 1 all shoes are by the back door so she must run barefoot.
Let be the probability that she runs barefoot on the morning of day . What conditions are satisfied in this problem which ensure exists? Show that its value is .
Find the expected number of days that will pass until the first morning that she finds all pairs of shoes at her front door.
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Paper 4, Section I, H
2012 commentLet be an irreducible Markov chain with . Define the meaning of the statements:
(i) state is transient,
(ii) state is aperiodic.
Give a criterion for transience that can be expressed in terms of the probabilities .
Prove that if a state is transient then all states are transient.
Prove that if a state is aperiodic then all states are aperiodic.
Suppose that unless is divisible by 3 . Given any other state , prove that unless is divisible by 3 .
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Paper 1, Section II, 20H
2012 commentA Markov chain has as its state space the integers, with
and otherwise. Assume .
Let if this is finite, and otherwise. Let be the total number of hits on 0 , and let be the total number of hits on 0 within times . Let
(i) Quoting an appropriate theorem, find, for every , the value of .
(ii) Show that if is any non-negative solution to the system of equations
then for all and .
(iii) Show that and .
(iv) Explain why for .
(v) Find for all .
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Paper 2, Section II, H
2012 commentLet be the symmetric random walk on vertices of a connected graph. At each step this walk jumps from the current vertex to a neighbouring vertex, choosing uniformly amongst them. Let . For each let and . Stating any theorems that you use:
(i) Prove that the invariant distribution satisfies detailed balance.
(ii) Use reversibility to explain why for all .
Consider a symmetric random walk on the graph shown below.

(iii) Find .
(iv) The removal of any edge leaves two disjoint components, one which includes and one which includes . Prove that , where is the number of edges in the component that contains .
(v) Show that for all .
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Paper 2, Section I, C
2012 commentUsing the method of characteristics, obtain a solution to the equation
subject to the Cauchy data for .
Sketch the characteristics and specify the greatest region of the plane in which a unique solution exists.
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Paper 4, Section I, D
2012 commentShow that the general solution of the wave equation
can be written in the form
Hence derive the solution subject to the initial conditions
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Paper 3, Section I, D
2012 commentFor the step-function
its convolution with itself is the hat-function
Find the Fourier transforms of and , and hence find the values of the integrals
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Paper 1, Section II, C
2012 commentConsider the regular Sturm-Liouville (S-L) system
where
with and for all in , and the boundary conditions on are
Show that with these boundary conditions, is self-adjoint. By considering , or otherwise, show that the eigenvalue can be written as
Now suppose that and , that and for all , and that and . Show that the eigenvalues of this regular S-L system are strictly positive. Assuming further that , solve the system explicitly, and with the aid of a graph, show that there exist infinitely many eigenvalues . Describe the behaviour of as .
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Paper 3, Section II, D
2012 commentConsider Legendre's equation
Show that if , with a non-negative integer, this equation has a solution , a polynomial of degree . Find and explicitly, subject to the condition .
The general solution of Laplace's equation in spherical polar coordinates, in the axisymmetric case, has the form
Hence, find the solution of Laplace's equation in the region satisfying the boundary conditions
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Paper 2, Section II, C
2012 commentConsider the linear differential operator defined by
on the interval . Given the boundary conditions and , find the Green's function for with these boundary conditions. Hence, or otherwise, obtain the solution of
subject to the above boundary conditions, where is a positive constant. Show that your piecewise solution is continuous at and has the value
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Paper 4, Section II, D
2012 commentLet be a two-dimensional domain with boundary , and let
where is a point in the interior of . From Green's second identity,
derive Green's third identity
[Here denotes the normal derivative on .]
Consider the Dirichlet problem on the unit :
Show that, with an appropriate function , the solution can be obtained by the formula
State the boundary conditions on and explain how is related to .
For , prove the identity
and deduce that if the point lies on the unit circle, then
Hence, using the method of images, or otherwise, find an expression for the function . [An expression for is not required.]
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Paper 3, Section I,
2012 commentDefine the notion of a connected component of a space .
If are connected subsets of such that , show that is connected.
Prove that any point is contained in a unique connected component.
Let consist of the points . What are the connected components of ?
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Paper 2, Section I,
2012 commentFor each case below, determine whether the given metrics and induce the same topology on . Justify your answers.
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Paper 1, Section II, F
2012 commentA topological space is said to be normal if each point of is a closed subset of and for each pair of closed sets with there are open sets so that and . In this case we say that the separate the .
Show that a compact Hausdorff space is normal. [Hint: first consider the case where is a point.]
For we define an equivalence relation on by for all , for . If and are pairwise disjoint closed subsets of a normal space , show that and may be separated by open subsets and such that . Deduce that the quotient space is also normal.
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Paper 4, Section II, F
2012 commentSuppose and are topological spaces. Define the product topology on . Let be the projection. Show that a map is continuous if and only if and are continuous.
Prove that if and are connected, then is connected.
Let be the topological space whose underlying set is , and whose open sets are of the form for , along with the empty set and the whole space. Describe the open sets in . Are the maps defined by and continuous? Justify your answers.
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Paper 1, Section I, D
2012 commentFind the LU factorization of the matrix and use it to solve the system via forward and backward substitution. [Other methods of solution are not acceptable.]
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Paper 4, Section I, D
2012 commentState the Dahlquist equivalence theorem regarding convergence of a multistep method.
The multistep method, with a real parameter ,
is of order 2 for any , and also of order 3 for . Determine all values of for which the method is convergent, and find the order of convergence.
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Paper 1, Section II, D
2012 commentFor a numerical method for solving , define the linear stability domain, and state when such a method is A-stable.
Determine all values of the real parameter for which the Runge-Kutta method
is A-stable.
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Paper 3, Section II, D
2012 commentDefine the QR factorization of an matrix and explain how it can be used to solve the least squares problem of finding the vector which minimises , where , and the norm is the Euclidean one.
Define a Householder transformation and show that it is an orthogonal matrix.
Using a Householder transformation, solve the least squares problem for
giving both and .
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Paper 2, Section II, D
2012 commentLet be the sequence of monic polynomials of degree orthogonal on the interval with respect to the weight function .
Prove that each has distinct zeros in the interval .
Let , and let satisfy the following three-term recurrence relation:
Set
Prove that , and deduce that all the eigenvalues of are distinct and reside in .
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Paper 1, Section I, 8H
2012 commentState the Lagrangian sufficiency theorem.
Use Lagrange multipliers to find the optimal values of and in the problem: maximize subject to and for all values of such that .
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Paper 2, Section I, H
2012 commentConsider the two-player zero-sum game with payoff matrix
Express the problem of finding the column player's optimal strategy as a linear programming problem in which is to be maximized subject to some constraints.
Solve this problem using the simplex algorithm and find the optimal strategy for the column player.
Find also, from the final tableau you obtain, both the value of the game and the row player's optimal strategy.
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Paper 4, Section II, 20H
2012 commentDescribe the Ford-Fulkerson algorithm.
State conditions under which the algorithm is guaranteed to terminate in a finite number of steps. Explain why it does so, and show that it finds a maximum flow. [You may assume that the value of a flow never exceeds the value of any cut.]
In a football league of teams the season is partly finished. Team has already won matches. Teams and are to meet in further matches. Thus the total number of remaining matches is . Assume there will be no drawn matches. We wish to determine whether it is possible for the outcomes of the remaining matches to occur in such a way that at the end of the season the numbers of wins by the teams are .
Invent a network flow problem in which the maximum flow from source to sink equals if and only if is a feasible vector of final wins.
Illustrate your idea by answering the question of whether or not is a possible profile of total end-of-season wins when , and with
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Paper 3, Section II, 21H
2012 commentFor given positive real numbers , consider the linear program
subject to for all for all ,
and for all .
(i) Consider the feasible solution in which and otherwise. Write down two basic feasible solutions of , one of which you can be sure is at least as good as . Are either of these basic feasible solutions of degenerate?
(ii) Starting from a general definition of a Lagrangian dual problem show that the dual of can be written as
What happens to the optimal value of this problem if the constraints and are removed?
Prove that is an optimal solution to if and only if there exist such that
[You may use any facts that you know from the general theory of linear programming provided that you state them.]
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Paper 4, Section I,
2012 commentIn terms of quantum states, what is meant by energy degeneracy?
A particle of mass is confined within the box and . The potential vanishes inside the box and is infinite outside. Find the allowed energies by considering a stationary state wavefunction of the form
Write down the normalised ground state wavefunction. Assuming that , give the energies of the first three excited states.
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Paper 3, Section I, C
2012 commentA one-dimensional quantum mechanical particle has normalised bound state energy eigenfunctions and corresponding non-degenerate energy eigenvalues . At the normalised wavefunction is given by
where and are real constants. Write down the expression for at a later time and give the probability that a measurement of the particle's energy will yield a value of .
Show that the expectation value of at time is given by
where .
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Paper 1, Section II, C
2012 commentShow that if the energy levels are discrete, the general solution of the Schrödinger equation
is a linear superposition of stationary states
where is a solution of the time-independent Schrödinger equation and are complex coefficients. Can this general solution be considered to be a stationary state? Justify your answer.
A linear operator acts on the orthonormal energy eigenfunctions as follows:
Obtain the eigenvalues of . Hence, find the normalised eigenfunctions of . In an experiment a measurement is made of at yielding an eigenvalue of 2 . What is the probability that a measurement at some later time will yield an eigenvalue of 2 ?
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Paper 3, Section II, C
2012 commentState the condition for a linear operator to be Hermitian.
Given the position and momentum operators and , define the angular momentum operators . Establish the commutation relations
and use these relations to show that is Hermitian assuming and are.
Consider a wavefunction of the form
where and is some constant. Show that is an eigenstate of the total angular momentum operator for all , and calculate the corresponding eigenvalue. For what values of is an eigenstate of ? What are the corresponding eigenvalues?
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Paper 2, Section II, C
2012 commentConsider a quantum mechanical particle in a one-dimensional potential , for which . Prove that when the energy eigenvalue is non-degenerate, the energy eigenfunction has definite parity.
Now assume the particle is in the double potential well
where and (U being large and positive). Obtain general expressions for the even parity energy eigenfunctions in terms of trigonometric and hyperbolic functions. Show that
where and .
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Paper 1, Section I, H
2012 commentDescribe the generalised likelihood ratio test and the type of statistical question for which it is useful.
Suppose that are independent and identically distributed random variables with the Gamma distribution, having density function . Similarly, are independent and identically distributed with the Gamma distribution. It is desired to test the hypothesis against . Derive the generalised likelihood ratio test and express it in terms of .
Let denote the value that a random variable having the distribution exceeds with probability . Explain how to decide the outcome of a size test when by knowing only the value of and the value , for some and , which you should specify.
[You may use the fact that the distribution is equivalent to the distribution.]
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Paper 2, Section I, H
2012 commentLet the sample have likelihood function . What does it mean to say is a sufficient statistic for ?
Show that if a certain factorization criterion is satisfied then is sufficient for .
Suppose that is sufficient for and there exist two samples, and , for which and does not depend on . Let
Show that is also sufficient for .
Explain why is not minimally sufficient for .
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Paper 4, Section II, H
2012 commentFrom each of 3 populations, data points are sampled and these are believed to obey
where , the are independent and identically distributed as , and is unknown. Let .
(i) Find expressions for and , the least squares estimates of and .
(ii) What are the distributions of and ?
(iii) Show that the residual sum of squares, , is given by
Calculate when ,
(iv) is the hypothesis that . Find an expression for the maximum likelihood estimator of under the assumption that is true. Calculate its value for the above data.
(v) Explain (stating without proof any relevant theory) the rationale for a statistic which can be referred to an distribution to test against the alternative that it is not true. What should be the degrees of freedom of this distribution? What would be the outcome of a size test of with the above data?
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Paper 1, Section II, H
2012 commentState and prove the Neyman-Pearson lemma.
A sample of two independent observations, , is taken from a distribution with density . It is desired to test against . Show that the best test of size can be expressed using the number such that
Is this the uniformly most powerful test of size for testing against
Suppose that the prior distribution of is , where . Find the test of against that minimizes the probability of error.
Let denote the power function of this test at . Show that
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Paper 4, Section II, I
2012 commentLet be a smooth projective curve of genus 2, defined over the complex numbers. Show that there is a morphism which is a double cover, ramified at six points.
Explain briefly why cannot be embedded into .
For any positive integer , show that there is a smooth affine plane curve which is a double cover of ramified at points.
[State clearly any theorems that you use.]
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Paper 3, Section II, I
2012 commentLet be the projective closure of the affine curve . Let denote the differential . Show that is smooth, and compute for all .
Calculate the genus of .
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Paper 2, Section II, I
2012 commentLet be a field, an ideal of , and let . Define the radical of and show that it is also an ideal.
The Nullstellensatz says that if is a maximal ideal, then the inclusion is an algebraic extension of fields. Suppose from now on that is algebraically closed. Assuming the above statement of the Nullstellensatz, prove the following.
(i) If is a maximal ideal, then , for some .
(ii) If , then , where
(iii) For an affine subvariety of , we set
Prove that for some affine subvariety , if and only if .
[Hint. Given , you may wish to consider the ideal in generated and .]
(iv) If is a finitely generated algebra over , and does not contain nilpotent elements, then there is an affine variety , for some , with .
Assuming , find when is the ideal in .
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Paper 1, Section II, I
2012 comment(a) Let be an affine variety, its ring of functions, and let . Assume is algebraically closed. Define the tangent space at . Prove the following assertions.
(i) A morphism of affine varieties induces a linear map
(ii) If and , then has the natural structure of an affine variety, and the natural morphism of into induces an isomorphism for all .
(iii) For all , the subset is a Zariski-closed subvariety of .
(b) Show that the set of nilpotent matrices
may be realised as an affine surface in , and determine its tangent space at all points .
Define what it means for two varieties and to be birationally equivalent, and show that the variety of nilpotent matrices is birationally equivalent to .
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Paper 3, Section II, G
2012 commentState the Mayer-Vietoris Theorem for a simplicial complex expressed as the union of two subcomplexes and . Explain briefly how the connecting homomorphism , which appears in the theorem, is defined. [You should include a proof that is well-defined, but need not verify that it is a homomorphism.]
Now suppose that , that is a solid torus , and that is the boundary torus of . Show that is an isomorphism, and hence calculate the homology groups of . [You may assume that a generator of may be represented by a 3 -cycle which is the sum of all the 3 -simplices of , with 'matching' orientations.]
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Paper 4, Section II, G
2012 commentState and prove the Lefschetz fixed-point theorem. Hence show that the -sphere does not admit a topological group structure for any even . [The existence and basic properties of simplicial homology with rational coefficients may be assumed.]
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Paper 2, Section II, G
2012 commentState the Seifert-Van Kampen Theorem. Deduce that if is a continuous map, where is path-connected, and is the space obtained by adjoining a disc to via , then is isomorphic to the quotient of by the smallest normal subgroup containing the image of .
State the classification theorem for connected triangulable 2-manifolds. Use the result of the previous paragraph to obtain a presentation of , where denotes the compact orientable 2 -manifold of genus .
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Paper 1, Section II, G
2012 commentDefine the notions of covering projection and of locally path-connected space. Show that a locally path-connected space is path-connected if it is connected.
Suppose and are continuous maps, the space is connected and locally path-connected and that is a covering projection. Suppose also that we are given base-points satisfying . Show that there is a continuous satisfying and if and only if the image of is contained in that of . [You may assume the path-lifting and homotopy-lifting properties of covering projections.]
Now suppose is locally path-connected, and both and are covering projections with connected domains. Show that and are homeomorphic as spaces over if and only if the images of their fundamental groups under and are conjugate subgroups of .
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Paper 2, Section II, E
2012 commentA solution of the -wave Schrödinger equation at large distances for a particle of mass with momentum and energy , has the form
Define the phase shift and verify that .
Write down a formula for the cross-section , for a particle of momentum scattering on a radially symmetric potential of finite range, as a function of the phase shifts for the partial waves with quantum number .
(i) Suppose that for . Show that there is a bound state of energy . Neglecting the contribution from partial waves with show that the cross section is
(ii) Suppose now that with and . Neglecting the contribution from partial waves with , derive an expression for the cross section , and show that it has a local maximum when . Discuss the interpretation of this phenomenon in terms of resonant behaviour and derive an expression for the decay width of the resonant state.
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Paper 4, Section II,
2012 commentConsider a one-dimensional crystal lattice of lattice spacing with the -th atom having position and momentum , for . The atoms interact with their nearest neighbours with a harmonic force and the classical Hamiltonian is
where we impose periodic boundary conditions: . Show that the normal mode frequencies for the classical harmonic vibrations of the system are given by
where , with integer and (for even, which you may assume) . What is the velocity of sound in this crystal?
Show how the system may be quantized to give the quantum operator
where and are creation and annihilation operators, respectively, whose commutation relations should be stated. Briefly describe the spectrum of energy eigenstates for this system, stating the definition of the ground state and giving the expression for the energy eigenvalue of any eigenstate.
The Debye-Waller factor associated with Bragg scattering from this crystal is defined by the matrix element
In the case where , calculate .
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Paper 3, Section II, E
2012 commentA simple model of a crystal consists of a 1D linear array of sites at positions , for all integer and separation , each occupied by a similar atom. The potential due to the atom at the origin is , which is symmetric: . The Hamiltonian, , for the atom at the -th site in isolation has electron eigenfunction with energy . Write down and state the relationship between and .
The Hamiltonian for an electron moving in the crystal is . Give an expression for .
In the tight-binding approximation for this model the are assumed to be orthonormal, , and the only non-zero matrix elements of and are
where . By considering the trial wavefunction , show that the time-dependent Schrödinger equation governing the amplitudes is
By examining a solution of the form
show that , the energy of the electron in the crystal, lies in a band given by
Using the fact that is a parity eigenstate show that
The electron in this model is now subject to an electric field in the direction of increasing , so that is replaced by , where is the charge on the electron. Assuming that , write down the new form of the time-dependent Schrödinger equation for the probability amplitudes . Verify that it has solutions of the form
where
Use this result to show that the dynamical behaviour of an electron near the bottom of an energy band is the same as that for a free particle in the presence of an electric field with an effective mass .
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Paper 1, Section II, E
2012 commentGive an account of the variational principle for establishing an upper bound on the ground-state energy of a particle moving in a potential in one dimension.
A particle of unit mass moves in the potential
with a positive constant. Explain why it is important that any trial wavefunction used to derive an upper bound on should be chosen to vanish for .
Use the trial wavefunction
where is a positive real parameter, to establish an upper bound for the energy of the ground state, and hence derive the lowest upper bound on as a function of .
Explain why the variational method cannot be used in this case to derive an upper bound for the energy of the first excited state.
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Paper 4, Section II,
2012 comment(a) Define the Moran model and Kingman's -coalescent. State and prove a theorem which describes the relationship between them. [You may use without proof a construction of the Moran model for all .]
(b) Let . Suppose that a population of individuals evolves according to the rules of the Moran model. Assume also that each individual in the population undergoes a mutation at constant rate . Each time a mutation occurs, we assume that the allelic type of the corresponding individual changes to an entirely new type, never seen before in the population. Let be the homozygosity probability, i.e., the probability that two individuals sampled without replacement from the population have the same genetic type. Give an expression for .
(c) Let denote the probability that a sample of size consists of one allelic type (monomorphic population). Show that , where denotes the sum of all the branch lengths in the genealogical tree of the sample - that is, , where is the first time that the genealogical tree of the sample has lineages. Deduce that
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Paper 3, Section II, K
2012 commentWe consider a system of two queues in tandem, as follows. Customers arrive in the first queue at rate . Each arriving customer is immediately served by one of infinitely many servers at rate . Immediately after service, customers join a single-server second queue which operates on a first-come, first-served basis, and has a service rate . After service in this second queue, each customer returns to the first queue with probability , and otherwise leaves the system forever. A schematic representation is given below:

(a) Let and denote the number of customers at time in queues number 1 and 2 respectively, including those currently in service at time . Give the transition rates of the Markov chain .
(b) Write down an equation satisfied by any invariant measure for this Markov chain. Let and . Define a measure by
Show that it is possible to find so that is an invariant measure of , if and only if . Give the values of and in this case.
(c) Assume now that . Show that the number of customers is not positive recurrent.
[Hint. One way to solve the problem is as follows. Assume it is positive recurrent. Observe that is greater than a queue with arrival rate . Deduce that is greater than a queue with arrival rate and service rate . You may use without proof the fact that the departure process from the first queue is then, at equilibrium, a Poisson process with rate , and you may use without proof properties of thinned Poisson processes.]
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Paper 2, Section II, K
2012 comment(a) A colony of bacteria evolves as follows. Let be a random variable with values in the positive integers. Each bacterium splits into copies of itself after an exponentially distributed time of parameter . Each of the daughters then splits in the same way but independently of everything else. This process keeps going forever. Let denote the number of bacteria at time . Specify the -matrix of the Markov chain . [It will be helpful to introduce , and you may assume for simplicity that
(b) Using the Kolmogorov forward equation, or otherwise, show that if , then for some to be explicitly determined in terms of . Assuming that , deduce the value of for all , and show that does not explode. [You may differentiate series term by term and exchange the order of summation without justification.]
(c) We now assume that with probability 1 . Fix and let . Show that satisfies
By making the change of variables , show that . Deduce that for all where .
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Paper 1, Section II,
2012 comment(a) Give the definition of a Poisson process with rate , using its transition rates. Show that for each , the distribution of is Poisson with a parameter to be specified.
Let and let denote the jump times of . What is the distribution of (You do not need to justify your answer.)
(b) Let . Compute the joint probability density function of given . Deduce that, given has the same distribution as the nondecreasing rearrangement of independent uniform random variables on .
(c) Starting from time 0, passengers arrive on platform at King's Cross station, with constant rate , in order to catch a train due to depart at time . Using the above results, or otherwise, find the expected total time waited by all passengers (the sum of all passengers' waiting times).
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Paper 4, Section II, B
2012 commentThe stationary Schrödinger equation in one dimension has the form
where can be assumed to be small. Using the Liouville-Green method, show that two approximate solutions in a region where are
where is suitably chosen.
Without deriving connection formulae in detail, describe how one obtains the condition
for the approximate energies of bound states in a smooth potential well. State the appropriate values of and .
Estimate the range of for which gives a good approximation to the true bound state energies in the cases
(i) ,
(ii) with small and positive,
(iii) with small and positive.
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Paper 3, Section II, B
2012 commentFind the two leading terms in the asymptotic expansion of the Laplace integral
as , where is smooth and positive on .
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Paper 1, Section II, B
2012 commentWhat precisely is meant by the statement that
as
Consider the Stieltjes integral
where is bounded and decays rapidly as , and . Find an asymptotic series for of the form , as , and prove that it has the asymptotic property.
In the case that , show that the coefficients satisfy the recurrence relation
and that . Hence find the first three terms in the asymptotic series.
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Paper 4, Section I, A
2012 commentConsider a one-dimensional dynamical system with generalized coordinate and momentum .
(a) Define the Poisson bracket of two functions and .
(b) Find the Poisson brackets and .
(c) Assuming Hamilton's equations of motion prove that
(d) State the condition for a transformation to be canonical in terms of the Poisson brackets found in (b). Use this to determine whether or not the following transformations are canonical:
(i) ,
(ii) ,
where is constant.
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Paper 3, Section I, A
2012 commentThe motion of a particle of charge and mass in an electromagnetic field with scalar potential and vector potential is characterized by the Lagrangian
(a) Show that the Euler-Lagrange equation is invariant under the gauge transformation
for an arbitrary function .
(b) Derive the equations of motion in terms of the electric and magnetic fields and .
[Recall that and .]
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Paper 2, Section I, A
2012 comment(a) The action for a system with a generalized coordinate is given by
State the Principle of Least Action and state the Euler-Lagrange equation.
(b) Consider a light rigid circular wire of radius and centre . The wire lies in a vertical plane, which rotates about the vertical axis through . At time the plane containing the wire makes an angle with a fixed vertical plane. A bead of mass is threaded onto the wire. The bead slides without friction along the wire, and its location is denoted by . The angle between the line and the downward vertical is .
Show that the Lagrangian of this system is
Calculate two independent constants of the motion, and explain their physical significance.
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Paper 1, Section I, A
2012 commentConsider a heavy symmetric top of mass , pinned at point , which is a distance from the centre of mass.
(a) Working in the body frame (where is the symmetry axis of the top) define the Euler angles and show that the components of the angular velocity can be expressed in terms of the Euler angles as
(b) Write down the Lagrangian of the top in terms of the Euler angles and the principal moments of inertia .
(c) Find the three constants of motion.
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Paper 4, Section II, A
2012 commentA homogenous thin rod of mass and length is constrained to rotate in a horizontal plane about its centre . A bead of mass is set to slide along the rod without friction. The bead is attracted to by a force resulting in a potential , where is the distance from .
(a) Identify suitable generalized coordinates and write down the Lagrangian of the system.
(b) Identify all conserved quantities.
(c) Derive the equations of motion and show that one of them can be written as
where the form of the effective potential should be found explicitly.
(d) Sketch the effective potential. Find and characterize all points of equilibrium.
(e) Find the frequencies of small oscillations around the stable equilibria.
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Paper 2, Section II, A
2012 commentConsider a rigid body with principal moments of inertia .
(a) Derive Euler's equations of torque-free motion
with components of the angular velocity given in the body frame.
(b) Show that rotation about the second principal axis is unstable if .
(c) The principal moments of inertia of a uniform cylinder of radius , height and mass about its centre of mass are
The cylinder has two identical cylindrical holes of radius drilled along its length. The axes of symmetry of the holes are at a distance from the axis of symmetry of the cylinder such that and . All three axes lie in a single plane. Compute the principal moments of inertia of the body.
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Paper 4, Section I, G
2012 commentDescribe the BB84 protocol for quantum key exchange.
Suppose we attempt to implement the BB84 protocol but cannot send single photons. Instead we send photons at a time all with the same polarization. An enemy can separate one of these photons from the other . Explain briefly how the enemy can intercept the key exchange without our knowledge.
Show that an enemy can find our common key if . Can she do so when (with suitable equipment)?
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Paper 3, Section , G
2012 commentDescribe the RSA system with public key and private key . Give a simple example of how the system is vulnerable to a homomorphism attack. Explain how a signature system prevents such an attack.
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Paper 2, Section I, G
2012 commentWhat is a (binary) linear code? What does it mean to say that a linear code has length and minimum weight ? When is a linear code perfect? Show that, if , there exists a perfect linear code of length and minimum weight 3 .
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Paper 1, Section I, G
2012 commentLet and be alphabets of sizes and a respectively. What does it mean to say that is a decodable code? State Kraft's inequality.
Suppose that a source emits letters from the alphabet , each letter occurring with (known) probability . Let be the codeword-length random variable for a decodable code , where . It is desired to find a decodable code that minimizes the expected value of . Establish the lower bound , and characterise when equality occurs. [Hint. You may use without proof the Cauchy-Schwarz inequality, that (for positive )
with equality if and only if for all i.]
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Paper 2, Section II, G
2012 commentWhat does it mean to say that is a linear feedback shift register? Let be a stream produced by such a register. Show that there exist with such that for all .
Describe and justify the Berlekamp-Massey method for 'breaking' a cipher stream arising from a linear feedback register of unknown length.
Let be three streams produced by linear feedback registers. Set
Show that is also a stream produced by a linear feedback register. Sketch proofs of any theorems you use.
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Paper 1, Section II, 12G
2012 commentDefine a cyclic binary code of length .
Show how codewords can be identified with polynomials in such a way that cyclic binary codes correspond to ideals in the polynomial ring with a suitably chosen multiplication rule.
Prove that any cyclic binary code has a unique generator, that is, a polynomial of minimum degree, such that the code consists of the multiples of this polynomial. Prove that the rank of the code equals , and show that divides .
Show that the repetition and parity check codes are cyclic, and determine their generators.
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Paper 4, Section I, E
2012 commentThe number density of a species of non-relativistic particles of mass , in equilibrium at temperature and chemical potential , is
where is the spin degeneracy. During primordial nucleosynthesis, deuterium, , forms through the nuclear reaction
where and are non-relativistic protons and neutrons. Write down the relationship between the chemical potentials in equilibrium.
Using the fact that , and explaining the approximations you make, show that
where is the deuterium binding energy, i.e. .
Let where is the baryon number density of the universe. Using the fact that , show that
where is the baryon asymmetry parameter
Briefly explain why primordial deuterium does not form until temperatures well below .
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Paper 3, Section I, E
2012 commentFor an ideal Fermi gas in equilibrium at temperature and chemical potential , the average occupation number of the th energy state, with energy , is
Discuss the limit . What is the Fermi energy How is it related to the Fermi momentum ? Explain why the density of states with momentum between and is proportional to and use this fact to deduce that the fermion number density at zero temperature takes the form
Consider an ideal Fermi gas that, at zero temperature, is either (i) non-relativistic or (ii) ultra-relativistic. In each case show that the fermion energy density takes the form
for some constant which you should compute.
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Paper 2, Section I, E
2012 commentThe Friedmann equation for the scale factor of a homogeneous and isotropic universe of mass density is
where and is a constant. The mass conservation equation for a fluid of mass density and pressure is
Conformal time is defined by . Show that
where . Hence show that the acceleration equation can be written as
Define the density parameter and show that in a matter-dominated era, in which , it satisfies the equation
Use this result to briefly explain the "flatness problem" of cosmology.
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Paper 1, Section I, E
2012 commentThe number density of photons in equilibrium at temperature is given by
where is Boltzmann's constant). Show that . Show further that , where is the photon energy density.
Write down the Friedmann equation for the scale factor of a flat homogeneous and isotropic universe. State the relation between and the mass density for a radiation-dominated universe and hence deduce the time-dependence of . How does the temperature depend on time?
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Paper 3, Section II, E
2012 commentIn a flat expanding universe with scale factor , average mass density and average pressure , the fractional density perturbations at co-moving wavenumber satisfy the equation
Discuss briefly the meaning of each term on the right hand side of this equation. What is the Jeans length , and what is its significance? How is it related to the Jeans mass?
How does the equation simplify at in a flat universe? Use your result to show that density perturbations can grow. For a growing density perturbation, how does compare to the inverse Hubble time?
Explain qualitatively why structure only forms after decoupling, and why cold dark matter is needed for structure formation.
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Paper 1, Section II, E
2012 commentThe Friedmann equation for the scale factor of a homogeneous and isotropic universe of mass density is
where . Explain how the value of the constant affects the late-time behaviour of .
Explain briefly why in a matter-dominated (zero-pressure) universe. By considering the scale factor of a closed universe as a function of conformal time , defined by , show that
where is the present density parameter, with . Use this result to show that
where is the present Hubble parameter. Find the time at which this model universe ends in a "big crunch".
Given that , obtain an expression for the present age of the universe in terms of and , according to this model. How does it compare with the age of a flat universe?
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Paper 4, Section II, I
2012 commentFor manifolds , define the terms tangent space to at a point and derivative of a smooth map . State the Inverse Function Theorem for smooth maps between manifolds without boundary.
Now let be a submanifold of and the inclusion map. By considering the map , or otherwise, show that is injective for each .
Show further that there exist local coordinates around and around such that is given in these coordinates by
where and . [You may assume that any open ball in is diffeomorphic to .]
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Paper 3, Section II, I
2012 commentFor a surface , define what is meant by the exponential mapping exp at , geodesic polar coordinates and geodesic circles.
Let be the coefficients of the first fundamental form in geodesic polar coordinates . Prove that and . Give an expression for the Gaussian curvature in terms of .
Prove that the Gaussian curvature at a point satisfies
where is the area of the region bounded by the geodesic circle of radius centred at .
[You may assume that and is an isometry. Taylor's theorem with any form of the remainder may be assumed if accurately stated.]
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Paper 2, Section II, I
2012 commentDefine the Gauss map for an oriented surface . Show that at each the derivative of the Gauss map
is self-adjoint. Define the principal curvatures of .
Now suppose that is compact (and without boundary). By considering the square of the distance to the origin, or otherwise, prove that has a point with .
[You may assume that the intersection of with a plane through the normal direction at contains a regular curve through .]
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Paper 1, Section II, I
2012 commentDefine the geodesic curvature of a regular curve in an oriented surface . When is along a curve?
Explain briefly what is meant by the Euler characteristic of a compact surface . State the global Gauss-Bonnet theorem with boundary terms.
Let be a surface with positive Gaussian curvature that is diffeomorphic to the sphere and let be two disjoint simple closed curves in . Can both and be geodesics? Can both and have constant geodesic curvature? Justify your answers.
[You may assume that the complement of a simple closed curve in consists of two open connected regions.]
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Paper 4, Section I, D
2012 commentDescribe the different types of bifurcation from steady states of a one-dimensional map of the form , and give examples of simple equations exhibiting each type.
Consider the map . What is the maximum value of for which the interval is mapped into itself?
Show that as increases from zero to its maximum value there is a saddle-node bifurcation and a period-doubling bifurcation, and determine the values of for which they occur.
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Paper 3, Section I, D
2012 commentState without proof Lyapunov's first theorem, carefully defining all the terms that you use.
Consider the dynamical system
By choosing a Lyapunov function , prove that the origin is asymptotically stable.
By factorising the expression for , or otherwise, show that the basin of attraction of the origin includes the set .
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Paper 2, Section I, D
2012 commentConsider the dynamical system
where is a constant.
(a) Show that there is a bifurcation from the fixed point at .
(b) Find the extended centre manifold at leading non-trivial order in . Hence find the type of bifurcation, paying particular attention to the special values and . [Hint. At leading order, the extended centre manifold is of the form , where are constants to be determined.]
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Paper 1, Section I,
2012 commentState the Poincaré-Bendixson theorem.
A model of a chemical process obeys the second-order system
where . Show that there is a unique fixed point at and that it is unstable if . Show that trajectories enter the region bounded by the lines , and , provided . Deduce that there is a periodic orbit when .
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Paper 4, Section II, D
2012 commentWhat is meant by the statement that a continuous map of an interval into itself has a horseshoe? State without proof the properties of such a map.
Define the property of chaos of such a map according to Glendinning.
A continuous map has a periodic orbit of period 5 , in which the elements satisfy and the points are visited in the order . Show that the map is chaotic. [The Intermediate Value theorem can be used without proof.]
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Paper 3, Section II, D
2012 commentConsider the dynamical system
(a) Show that the fixed point at the origin is an unstable node or focus, and that the fixed point at is a saddle point.
(b) By considering the phase plane , or otherwise, show graphically that the maximum value of for any periodic orbit is less than one.
(c) By writing the system in terms of the variables and , or otherwise, show that for any periodic orbit
Deduce that if there are no periodic orbits.
(d) If the system (1) is Hamiltonian and has homoclinic orbit
which approaches as . Now suppose that are very small and that we seek the value of corresponding to a periodic orbit very close to . By using equation (3) in equation (2), find an approximation to the largest value of for a periodic orbit when are very small.
[Hint. You may use the fact that
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Paper 4, Section II, B
2012 commentThe charge and current densities are given by and respectively. The electromagnetic scalar and vector potentials are given by and respectively. Explain how one can regard as a four-vector that obeys the current conservation rule .
In the Lorenz gauge , derive the wave equation that relates to and hence show that it is consistent to treat as a four-vector.
In the Lorenz gauge, with , a plane wave solution for is given by
where and are four-vectors with
Show that .
Interpret the components of in terms of the frequency and wavelength of the wave.
Find what residual gauge freedom there is and use it to show that it is possible to set . What then is the physical meaning of the components of ?
An observer at rest in a frame measures the angular frequency of a plane wave travelling parallel to the -axis to be . A second observer travelling at velocity in parallel to the -axis measures the radiation to have frequency . Express in terms of .
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Paper 3, Section II, B
2012 commentThe non-relativistic Larmor formula for the power, , radiated by a particle of charge and mass that is being accelerated with an acceleration a is
Starting from the Liénard-Wiechert potentials, sketch a derivation of this result. Explain briefly why the relativistic generalization of this formula is
where is the relativistic momentum of the particle and is the proper time along the worldline of the particle.
A particle of mass and charge moves in a plane perpendicular to a constant magnetic field . At time as seen by an observer at rest, the particle has energy . At what rate is electromagnetic energy radiated by this particle?
At time according to the observer , the particle has energy . Find an expression for in terms of and .
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Paper 1, Section II, B
2012 commentA particle of mass and charge moves relativistically under the influence of a constant electric field in the positive -direction, and a constant magnetic field also in the positive -direction.
In some inertial observer's coordinate system, the particle starts at
with velocity given by
where the dot indicates differentiation with respect to the proper time of the particle. Show that the subsequent motion of the particle, as seen by the inertial observer, is a helix.
a) What is the radius of the helix as seen by the inertial observer?
b) What are the and coordinates of the axis of the helix?
c) What is the coordinate of the particle after a proper time has elapsed, as measured by the particle?
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Paper 4, Section II, C
2012 commentA steady, two-dimensional flow in the region takes the form at large , where is a positive constant. The boundary at is rigid and no-slip. Consider the velocity field with stream function , where and and is the kinematic viscosity. Show that this velocity field satisfies the Navier-Stokes equations provided that satisfies
What are the conditions on at and as ?
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Paper 2, Section II, C
2012 commentAn incompressible viscous liquid occupies the long thin region for , where with and . The top boundary at is rigid and stationary. The bottom boundary at is rigid and moving at velocity . Fluid can move in and out of the ends and , where the pressure is the same, namely .
Explaining the approximations of lubrication theory as you use them, find the velocity profile in the long thin region, and show that the volume flux (per unit width in the -direction) is
Find also the value of (i) where the pressure is maximum, (ii) where the tangential viscous stress on the bottom vanishes, and (iii) where the tangential viscous stress on the top vanishes.
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Paper 3, Section II, C
2012 commentFor two Stokes flows and inside the same volume with different boundary conditions on its boundary , prove the reciprocal theorem
where and are the stress fields associated with the flows.
When a rigid sphere of radius translates with velocity through unbounded fluid at rest at infinity, it may be shown that the traction per unit area, , exerted by the sphere on the fluid has the uniform value over the sphere surface. Find the drag on the sphere.
Suppose that the same sphere is now free of external forces and is placed with its centre at the origin in an unbounded Stokes flow given in the absence of the sphere as . By applying the reciprocal theorem to the perturbation to the flow generated by the presence of the sphere, and assuming this tends to zero sufficiently rapidly at infinity, show that the instantaneous velocity of the centre of the sphere is
where the integral is taken over the sphere of radius .
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Paper 1, Section II, C
2012 commentDefine the strain-rate tensor in terms of the velocity components . Write down the relation between , the pressure and the stress in an incompressible Newtonian fluid of viscosity . Show that the local rate of stress-working is equal to the local rate of dissipation .
An incompressible fluid of density and viscosity occupies the semi-infinite region above a rigid plane boundary which oscillates with velocity . The fluid is at rest at infinity. Determine the velocity field produced by the boundary motion after any transients have decayed.
Show that the time-averaged rate of dissipation is
per unit area of the boundary. Verify that this is equal to the time average of the rate of working by the boundary on the fluid per unit area.
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Paper 4, Section I, E
2012 commentUse the Laplace kernel method to write integral representations in the complex -plane for two linearly independent solutions of the confluent hypergeometric equation
in the case that and are not integers.
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Paper 3, Section I, E
2012 commentThe Beta function, denoted by , is defined by
where denotes the Gamma function. It can be shown that
By computing this integral for the particular case of , and by employing analytic continuation, deduce that satisfies the functional equation
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Paper 2, Section I, E
2012 commentThe hypergeometric function is defined as the particular solution of the second order linear ODE characterised by the Papperitz symbol
that is analytic at and satisfies .
Using the fact that a second solution of the above ODE is of the form
where is analytic in the neighbourhood of the origin, express in terms of .
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Paper 1, Section I, E
2012 commentRecall that if is analytic in a neighbourhood of , then
where is the real part of . Use this fact to construct the imaginary part of an analytic function whose real part is given by
where is real and has sufficient smoothness and decay.
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Paper 2, Section II,
2012 commentLet the complex function satisfy
where is a positive constant. The unified transform method implies that the solution of any well-posed problem for the above equation is given by
where is the union of the rays and denotes the Fourier transform of the initial condition , and denote the -transforms of the boundary values :
Furthermore, and are related via the so-called global relation
where denotes the Fourier transform of .
(a) Assuming the validity of (1) and (2), use the global relation to eliminate from equation (1).
(b) For the particular case that
where and are real numbers, use the representation obtained in (a) to express the solution in terms of an integral along the real axis and an integral along (you should not attempt to evaluate these integrals). Show that it is possible to deform these two integrals to a single integral along a new contour , which you should sketch.
[You may assume the validity of Jordan's lemma.]
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Paper 1, Section II, E
2012 comment(a) Suppose that , is analytic in the upper-half complex -plane and as . Show that the real and imaginary parts of , denoted by and respectively, satisfy the so-called Kramers-Kronig formulae:
Here, denotes the Hilbert transform, i.e.,
where denotes the principal value integral.
(b) Let the real function satisfy the Laplace equation in the upper-half complex z-plane, i.e.,
Assuming that decays for large and for large , show that is an analytic function for . Then, find an expression for in terms of .
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Paper 4, Section II, H
2012 commentLet be a field of rational functions in variables over , and let be the elementary symmetric polynomials:
and let be the subfield of generated by . Let , and . Let be the subfield of generated by over . Find the degree .
[Standard facts about the fields and Galois extensions can be quoted without proof, as long as they are clearly stated.]
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Paper 3, Section II, H
2012 commentLet be a power of the prime , and be a finite field consisting of elements.
Let be a positive integer prime to , and be the cyclotomic extension obtained by adjoining all th roots of unity to . Prove that is a finite field with elements, where is the order of the element in the multiplicative group of the .
Explain why what is proven above specialises to the following fact: the finite field for an odd prime contains a square root of if and only if .
[Standard facts on finite fields and their extensions can be quoted without proof, as long as they are clearly stated.]
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Paper 2, Section II, H
2012 commentLet be subfields of with .
Suppose that is contained in and is a finite Galois extension of odd degree. Prove that is also contained in .
Give one concrete example of as above with . Also give an example in which is contained in and has odd degree, but is not Galois and is not contained in .
[Standard facts on fields and their extensions can be quoted without proof, as long as they are clearly stated.]
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Paper 1, Section II, 18H
2012 commentList all subfields of the cyclotomic field obtained by adjoining all 20 th roots of unity to , and draw the lattice diagram of inclusions among them. Write all the subfields in the form or . Briefly justify your answer.
[The description of the Galois group of cyclotomic fields and the fundamental theorem of Galois theory can be used freely without proof.]
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Paper 4, Section II, B
2012 commentThe metric for a homogenous isotropic universe, in comoving coordinates, can be written as
where and are some functions.
Write down expressions for the Hubble parameter and the deceleration parameter in terms of and , where is conformal time, defined by .
The universe is composed of a perfect fluid of density and pressure , where is a constant. Defining , where , show that
where is the curvature parameter or and . Hence deduce that
and
where is a constant. Given that , sketch curves of against in the case when .
[You may assume an Einstein equation, for the given metric, in the form
and the energy conservation equation
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Paper 2, Section II, B
2012 commentThe metric of any two-dimensional rotationally-symmetric curved space can be written in terms of polar coordinates, , with , as
where . Show that the Christoffel symbols and are each zero, and compute and .
The Ricci tensor is defined by
where a comma here denotes partial derivative. Prove that and that
Suppose now that, in this space, the Ricci scalar takes the constant value . Find a differential equation for .
By a suitable coordinate transformation unchanged, this space of constant Ricci scalar can be described by the metric
From this coordinate transformation, find and in terms of . Deduce that
where , and is a positive constant.
[You may use
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Paper 3, Section II, 37B
2012 comment(i) The Schwarzschild metric is given by
Consider a time-like geodesic , where is the proper time, lying in the plane . Use the Lagrangian to derive the equations governing the geodesic, showing that
with constant, and hence demonstrate that
where . State which term in this equation makes it different from an analogous equation in Newtonian theory.
(ii) Now consider Kruskal coordinates, in which the Schwarzschild and are replaced by and , defined for by
and for by
Given that the metric in these coordinates is
where is defined implicitly by
sketch the Kruskal diagram, indicating the positions of the singularity at , the event horizon at , and general lines of constant and of constant .
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Paper 1, Section II, B
2012 comment(i) Using the condition that the metric tensor is covariantly constant, derive an expression for the Christoffel symbol .
(ii) Show that
Hence establish the covariant divergence formula
where is the determinant of the metric tensor.
[It may be assumed that for any invertible matrix ].
(iii) The Kerr-Newman metric, describing the spacetime outside a rotating black hole of mass , charge and angular momentum per unit mass , is given in appropriate units by
where and . Explain why this metric is stationary, and make a choice of one of the parameters which reduces it to a static metric.
Show that, in the static metric obtained, the equation
for a function admits solutions of the form
where is constant and satisfies an ordinary differential equation which should be found.
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Paper 4, Section I, G
2012 commentExplain briefly how to extend a Möbius transformation
from the boundary of the upper half-space to give a hyperbolic isometry of the upper half-space. Write down explicitly the extension of the transformation for any constant .
Show that, if has an axis, which is a hyperbolic line that is mapped onto itself by with the orientation preserved, then moves each point of this axis by the same hyperbolic distance, say. Prove that
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Paper 3, Section I, G
2012 commentLet be a Möbius transformation acting on the Riemann sphere. Show that, if is not loxodromic, then there is a disc in the Riemann sphere with . Describe all such discs for each Möbius transformation .
Hence, or otherwise, show that the group of Möbius transformations generated by
does not map any disc onto itself.
Describe the set of points of the Riemann sphere at which acts discontinuously. What is the quotient of this set by the action of ?
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Paper 2, Section I, G
2012 commentDefine the modular group acting on the upper half-plane. Explain briefly why it acts discontinuously and describe a fundamental domain. You should prove that the region which you describe is a fundamental domain.
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Paper 1, Section I, G
2012 commentLet be a crystallographic group of the Euclidean plane. Define the lattice and the point group of . Suppose that the lattice for is . Show that there are five different possibilities for the point group. Show that at least one of these point groups can arise from two groups that are not conjugate in the group of all isometries of the Euclidean plane.
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Paper 1, Section II, G
2012 commentDefine the axis of a loxodromic Möbius transformation acting on hyperbolic 3-space.
When do two loxodromic transformations commute? Justify your answer.
Let be a Kleinian group that contains a loxodromic transformation. Show that the fixed point of any loxodromic transformation in lies in the limit set of . Prove that the set of such fixed points is dense in the limit set. Give examples to show that the set of such fixed points can be equal to the limit set or a proper subset.
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Paper 4, Section II,
2012 commentDefine the Hausdorff dimension of a subset of the Euclidean plane.
Let be a closed disc of radius in the Euclidean plane. Define a sequence of sets , as follows: and for each a subset is produced by replacing each component disc of by three disjoint, closed discs inside with radius at most times the radius of . Let be the intersection of the sets . Show that if the factors converge to a limit with , then the Hausdorff dimension of is at most .
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Paper 4, Section II, F
2012 comment(a) Show that every finite tree of order at least 2 has a leaf. Hence, or otherwise, show that a tree of order must have precisely edges.
(b) Let be a graph. Explain briefly why .
Let , and assume . By induction on , or otherwise, show that has a subgraph with . Hence, or otherwise, show that if is a tree of order then .
(c) Let be integers, let and let be a tree of order . Show that whenever the edges of the complete graph are coloured blue and yellow then it must contain either a blue or a yellow .
Does this remain true if is replaced by ? Justify your answer.
[The independence number of a graph is the size of the largest set of vertices such that no edge of joins two points of . Recall that is the chromatic number and are respectively the minimal/maximal degrees of vertices in .]
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Paper 3, Section II,
2012 commentLet be a graph with at least one edge. Define ex , where is an integer with . Without assuming the Erdős-Stone theorem, show that the sequence converges as .
State precisely the Erdős-Stone theorem. Hence determine, with justification,
Let be another graph with at least one edge. For each integer such that , let
and let
Find, with justification, and .
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Paper 2, Section II, F
2012 commentLet be a -connected graph . Let and let with . Show that contains paths from to with any two having only the vertex in common.
[No form of Menger's theorem or of the Max-Flow-Min-Cut theorem may be assumed without proof.]
Deduce that must contain a cycle of length at least .
Suppose further that has no independent set of vertices of size . Show that is Hamiltonian.
[Hint. If not, let be a cycle of maximum length in and let ; consider the set of vertices on immediately preceding the endvertices of a collection of paths from to that have only the vertex in common.]
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Paper 1, Section II, F
2012 commentState Markov's inequality and Chebyshev's inequality.
Let denote the probability space of bipartite graphs with vertex classes and , with each possible edge , present, independently, with probability . Let be the number of subgraphs of that are isomorphic to the complete bipartite graph . Write down and . Hence show that is a threshold for to contain , in the sense that if then a. e. contains a , whereas if then a. e. does not contain a .
By modifying a random for suitably chosen , show that, for each , there exists a bipartite graph with vertices in each class such that but
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Paper 3, Section II, D
2012 commentConsider a one-parameter group of transformations acting on
where is a group parameter and are constants.
(a) Find a vector field which generates this group.
(b) Find two independent Lie point symmetries and of the
which are of the form (1).
(c) Find three functionally-independent invariants of , and do the same for . Find a non-constant function which is invariant under both and .
(d) Explain why all the solutions of (2) that are invariant under a two-parameter group of transformations generated by vector fields
are of the form , where is a function of one variable. Find an ODE for characterising these group-invariant solutions.
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Paper 2, Section II, D
2012 commentConsider the equation for the function
(a) Write equation (1) in the Hamiltonian form
where the functional should be given. Use equation (1), together with the boundary conditions and as , to show that is independent of .
(b) Use the Gelfand-Levitan-Marchenko equation
to find the one soliton solution of the KdV equation, i.e.
[Hint. Consider , with , where are constants, and should be regarded as a parameter in equation (2). You may use any facts about the Inverse Scattering Transform without proof.]
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Paper 1, Section II, D
2012 commentState the Arnold-Liouville theorem.
Consider an integrable system with six-dimensional phase space, and assume that on any Liouville tori , where .
(a) Define the action variables and use Stokes' theorem to show that the actions are independent of the choice of the cycles.
(b) Define the generating function, and show that the angle coordinates are periodic with period .
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Paper 3, Section II, G
2012 commentState the closed graph theorem.
(i) Let be a Banach space and a vector space. Suppose that is endowed with two norms and and that there is a constant such that for all . Suppose that is a Banach space with respect to both norms. Suppose that is a linear operator, and that it is bounded when is endowed with the norm. Show that it is also bounded when is endowed with the norm.
(ii) Suppose that is a normed space and that is a sequence with for all in the dual space . Show that there is an such that
for all .
(iii) Suppose that is the space of bounded continuous functions with the sup norm, and that is the subspace of continuously differentiable functions with bounded derivative. Let be defined by . Show that the graph of is closed, but that is not bounded.
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Paper 4, Section II, G
2012 commentLet be a Banach space and suppose that is a bounded linear operator. What is an eigenvalue of What is the spectrum of
Let be the space of continuous real-valued functions endowed with the sup norm. Define an operator by
where
Prove the following facts about :
(i) and the second derivative is equal to for ;
(ii) is compact;
(iii) has infinitely many eigenvalues;
(iv) 0 is not an eigenvalue of ;
(v) .
[The Arzelà-Ascoli theorem may be assumed without proof.]
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Paper 2, Section II, G
2012 commentWhat is meant by a normal topological space? State and prove Urysohn's lemma.
Let be a normal topological space and let be closed. Show that there is a continuous function with if, and only if, is a countable intersection of open sets.
[Hint. If then consider , where the functions are supplied by an appropriate application of Urysohn's lemma.]
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Paper 1, Section II, G
2012 commentWhat is meant by the dual of a normed space ? Show that is a Banach space.
Let , the space of functions possessing a bounded, continuous first derivative. Endow with the sup norm . Which of the following maps are elements of ? Give brief justifications or counterexamples as appropriate.
- ;
- .
Now suppose that is a (real) Hilbert space. State and prove the Riesz representation theorem. Describe the natural and show that it is surjective.
[All normed spaces are over . You may assume that if is a closed subspace of a Hilbert space then
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Paper 2, Section II, H
2012 commentExplain what is meant by a substructure of a -structure , where is a first-order signature (possibly including both predicate symbols and function symbols). Show that if is a substructure of , and is a first-order formula over with free variables, then if is quantifier-free. Show also that if is an existential formula (that is, one of the form where is quantifier-free), and if is a universal formula. Give examples to show that the two latter inclusions can be strict.
Show also that
(a) if is a first-order theory whose axioms are all universal sentences, then any substructure of a -model is a -model;
(b) if is a first-order theory such that every first-order formula is -provably equivalent to a universal formula (that is, for some universal ), and is a sub-T-model of a -model , then for every first-order formula with free variables.
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Paper 4, Section II, H
2012 commentState and prove Hartogs' lemma. [You may assume the result that any well-ordered set is isomorphic to a unique ordinal.]
Let and be sets such that there is a bijection . Show, without assuming the Axiom of Choice, that there is either a surjection or an injection . By setting (the Hartogs ordinal of ) and considering , show that the assertion 'For all infinite cardinals , we have implies the Axiom of Choice. [You may assume the Cantor-Bernstein theorem.]
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Paper 3, Section II, H
2012 commentWrite down either the synthetic or the recursive definitions of ordinal addition and multiplication. Using your definitions, give proofs or counterexamples for the following statements:
(i) For all and , we have .
(ii) For all and , we have .
(iii) For all and with , there exist and with and .
(iv) For all and with , there exist and with and .
(v) For every , either there exists a cofinal map (that is, one such that , or there exists such that
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Paper 1, Section II, H
2012 commentState Zorn's lemma, and show how it may be deduced from the Axiom of Choice using the Bourbaki-Witt theorem (which should be clearly stated but not proved).
Show that, if and are distinct elements of a distributive lattice , there is a lattice homomorphism with . Indicate briefly how this result may be used to prove the completeness theorem for propositional logic.
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Paper 4, Section I,
2012 commentThe master equation describing the evolution of the probability that a population has members at time takes the form
where the functions and are both positive for all .
From (1) derive the corresponding Fokker-Planck equation in the form
making clear any assumptions that you make and giving explicit forms for and .
Assume that (2) has a steady state solution and that is a decreasing function of with a single zero at . Under what circumstances may be approximated by a Gaussian centred at and what is the corresponding estimate of the variance?
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Paper 3, Section I, C
2012 commentConsider a model of insect dispersal in two dimensions given by
where is a radial coordinate, is time, is the density of insects and is a constant coefficient such that is a diffusivity.
Show that under suitable assumptions
where is constant, and interpret this condition.
Suppose that after a long time the form of depends only on and (and is thus independent of any detailed form of the initial condition). Show that there is a solution of the form
and deduce that the function satisfies
Show that this equation has a continuous solution with for and for , and determine . Hence determine the area within which as a function of .
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Paper 2, Section I,
2012 commentConsider a birth-death process in which the birth rate per individual is and the death rate per individual in a population of size is .
Let be the probability that the population has size at time . Write down the master equation for the system, giving an expression for .
Show that
where denotes the mean.
Deduce that in a steady state .
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Paper 1, Section I,
2012 commentKrill is the main food source for baleen whales. The following model has been proposed for the coupled evolution of populations of krill and whales, with being the number of krill and being the number of whales:
where and are positive constants.
Give a biological interpretation for the form of the two differential equations.
Show that a steady state is possible with and and write down expressions for the steady-state values of and .
Determine whether this steady state is stable.
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Paper 3, Section II, C
2012 commentConsider the two-variable reaction-diffusion system
where and are positive constants.
Show that there is one possible spatially homogeneous steady state with and and show that it is stable to small-amplitude spatially homogeneous disturbances provided that , where
Now assuming that the condition is satisfied, investigate the stability of the homogeneous steady state to spatially varying perturbations by considering the timedependence of disturbances whose spatial form is such that and , with constant. Show that such disturbances vary as , where is one of the roots of
By comparison with the stability condition for the homogeneous case above, give a simple argument as to why the system must be stable if .
Show that the boundary between stability and instability (as some combination of and is varied) must correspond to .
Deduce that is a necessary condition for instability and, furthermore, that instability will occur for some if
Deduce that the value of at which instability occurs as the stability boundary is crossed is given by
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Paper 2, Section II, C
2012 commentA population of blowflies is modelled by the equation
where is a constant death rate and is a function of one variable such that for , with as and as . The constants and are all positive, with . Give a brief biological motivation for the term , in which you explain both the form of the function and the appearance of a delay time .
A suitable model for is , where is a positive constant. Show that in this case there is a single steady state of the system with non-zero population, i.e. with , with constant.
Now consider the stability of this steady state. Show that if , with small, then satisfies a delay differential equation of the form
where is a constant to be determined. Show that is a solution of (2) if . If , where and are both real, write down two equations relating and .
Deduce that the steady state is stable if . Show that, for this particular model for is possible only if .
By considering decreasing from small negative values, show that an instability will appear when , where .
Deduce that the steady state of (1) is unstable if
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Paper 4, Section II, F
2012 commentLet where and are distinct primes with . By computing the relative traces where runs through the three quadratic subfields of , show that the algebraic integers in have the form
where are rational integers. Show further that if and are both even then and are both even. Hence prove that an integral basis for is
Calculate the discriminant of .
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Paper 2, Section II, F
2012 commentLet where is a root of . Factor the elements 2,3 , and as products of prime ideals in . Hence compute the class group of .
Show that the equation has no integer solutions.
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Paper 1, Section II, F
2012 commentLet be a number field, and its ring of integers. Write down a characterisation of the units in in terms of the norm. Without assuming Dirichlet's units theorem, prove that for a quadratic field the quotient of the unit group by is cyclic (i.e. generated by one element). Find a generator in the cases and .
Determine all integer solutions of the equation for .
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Paper 4, Section I, I
2012 commentDefine what it means for the composite natural number to be a pseudoprime to the base .
Find the number of bases (less than 21) to which 21 is a pseudoprime. [You may, if you wish, assume the Chinese Remainder Theorem.]
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Paper 3, Section I, I
2012 commentDefine the discriminant of the binary quadratic form .
Assuming that this form is positive definite, define what it means for to be reduced.
Show that there are precisely two reduced positive definite binary quadratic forms of discriminant .
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Paper 2, Section I, I
2012 commentDefine the Legendre symbol and the Jacobi symbol.
State the law of quadratic reciprocity for the Jacobi symbol.
Compute the value of the Jacobi symbol , stating clearly any results you use.
-
Paper 1, Section I, I
2012 commentShow that the continued fraction for is .
Hence, or otherwise, find a solution to the equation in positive integers and . Write down an expression for another solution.
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Paper 4, Section II, I
2012 commentLet be a function, where denotes the (positive) natural numbers.
Define what it means for to be a multiplicative function.
Prove that if is a multiplicative function, then the function defined by
is also multiplicative.
Define the Möbius function . Is multiplicative? Briefly justify your answer.
Compute
for all positive integers .
Define the Riemann zeta function for complex numbers with .
Prove that if is a complex number with , then
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Paper 3 , Section II, I
2012 commentLet be an odd prime. Prove that the multiplicative groups are cyclic for . [You may assume that the multiplicative group is cyclic.]
Find an integer which generates for all , justifying your answer.
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Paper 4, Section II,
2012 comment(i) Formulate the conjugate gradient method for the solution of a system with and .
(ii) Prove that if the conjugate gradient method is applied in exact arithmetic then, for any , termination occurs after at most iterations.
(iii) The polynomial is the minimal polynomial of the matrix if it is the polynomial of lowest degree that satisfies Note that .] Give an example of a symmetric positive definite matrix with a quadratic minimal polynomial.
Prove that (in exact arithmetic) the conjugate gradient method requires at most iterations to calculate the exact solution of , where is the degree of the minimal polynomial of .
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Paper 2, Section II, D
2012 comment(i) The diffusion equation
with the initial condition , and with zero boundary conditions at and , can be solved numerically by the method
where , and . Prove that implies convergence.
(ii) By discretising the diffusion equation and employing the same notation as in (i) above, determine [without using Fourier analysis] conditions on and the constant such that the method
is stable.
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Paper 3, Section II, D
2012 commentThe inverse discrete Fourier transform is given by the formula
Here, is the primitive root of unity of degree and
(i) Show how to assemble in a small number of operations if the Fourier transforms of the even and odd parts of ,
are already known.
(ii) Describe the Fast Fourier Transform (FFT) method for evaluating , and draw a relevant diagram for .
(iii) Find the costs of the FFT method for (only multiplications count).
(iv) For use the FFT method to find when:
(a) ,
(b) .
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Paper 1, Section II, D
2012 commentThe Poisson equation in the unit interval on is discretised with the formula
where and are the grid points.
(i) Define the above system of equations in vector form and describe the relaxed Jacobi method with relaxation parameter for solving this linear system. For and being the exact solution and the iterated solution respectively, let be the error and the iteration matrix, so that
Express in terms of the matrix , the diagonal part of and , and find the eigenvectors and the eigenvalues of .
(ii) For as above, let
be the expansion of the error with respect to the eigenvectors of . Derive conditions on such that the method converges for any , and prove that, for any such , the rate of convergence of is not faster than .
(iii) Show that, for some , the high frequency components of the error tend to zero much faster than . Determine the optimal parameter which provides the largest suppression of the high frequency components per iteration, and find the corresponding attenuation factor (i.e., the least such that for .
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Paper 3, Section II, J
2012 commentA state variable is subject to dynamics
where is a scalar control variable constrained to the interval . Given an initial value , let denote the minimal time required to bring the state to . Prove that
Explain how this equation figures in Pontryagin's maximum principle.
Use Pontryagin's maximum principle to show that, on an optimal trajectory, only takes the values 1 and , and that it makes at most one switch between them.
Show that is optimal when .
Find the optimal control when .
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Paper 4, Section II, J
2012 commentA factory has a tank of capacity in which it stores chemical waste. Each week the factory produces, independently of other weeks, an amount of waste that is equally likely to be 0,1 , or . If the amount of waste exceeds the remaining space in the tank then the excess must be specially handled at a cost of per . The tank may be emptied or not at the end of each week. Emptying costs , plus a variable cost of for each of its content. It is always emptied when it ends the week full.
It is desired to minimize the average cost per week. Write down equations from which one can determine when it is optimal to empty the tank.
Find the average cost per week of a policy , which empties the tank if and only if its content at the end of the week is 2 or .
Describe the policy improvement algorithm. Explain why, starting from , this algorithm will find an optimal policy in at most three iterations.
Prove that is optimal if and only if .
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Paper 2, Section II, J
2012 commentDescribe the elements of a generic stochastic dynamic programming equation for the problem of maximizing the expected sum of discounted rewards accrued at times What is meant by the positive case? What is specially true in this case that is not true in general?
An investor owns a single asset which he may sell once, on any of the days . On day he will be offered a price . This value is unknown until day , is independent of all other offers, and a priori it is uniformly distributed on . Offers remain open, so that on day he may sell the asset for the best of the offers made on days . If he sells for on day then the reward is . Show from first principles that if then there exists such that the expected reward is maximized by selling the first day the offer is at least .
For , find both and the expected reward under the optimal policy.
Explain what is special about the case .
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Paper 4, Section II, 30B
2012 commenti) State the Lax-Milgram lemma.
ii) Consider the boundary value problem
where is a bounded domain in with a smooth boundary, is the exterior unit normal vector to , and . Show (using the Lax-Milgram lemma) that the boundary value problem has a unique weak solution in the space
[Hint. Show that
and then use the fact that is dense in
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Paper 3, Section II, 30B
2012 commentConsider the nonlinear partial differential equation for a function ,
where .
(i) Find a transformation such that satisfies the heat equation
if (1) holds for .
(ii) Use the transformation obtained in (i) (and its inverse) to find a solution to the initial value problem (1), (2).
[Hint. Use the fundamental solution of the heat equation.]
(iii) The equation (1) is posed on a bounded domain with smooth boundary, subject to the initial condition (2) on and inhomogeneous Dirichlet boundary conditions
where is a bounded function. Use the maximum-minimum principle to prove that there exists at most one classical solution of this boundary value problem.
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Paper 1, Section II, B
2012 commentLet for all . Consider the partial differential equation for ,
subject to the Cauchy condition .
i) Compute the solution of the Cauchy problem by the method of characteristics.
ii) Prove that the domain of definition of the solution contains
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Paper 2, Section II, B
2012 commentConsider the elliptic Dirichlet problem on bounded with a smooth boundary:
Assume that and .
(i) State the strong Minimum-Maximum Principle for uniformly elliptic operators.
(ii) Prove that there exists at most one classical solution of the boundary value problem.
(iii) Assuming further that in , use the maximum principle to obtain an upper bound on the solution (assuming that it exists).
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Paper 4, Section II, A
2012 commentSetting , the raising and lowering operators for angular momentum satisfy
where . Find the matrix representation for in the basis

Suppose that the angular momentum of the state is measured in the direction to be . Find the components of , expressing each component by a single term consisting of products of powers of and multiplied by constants.
Suppose that two measurements of a total angular momentum 1 system are made. The first is made in the third direction with value , and the second measurement is subsequently immediately made in direction . What is the probability that the second measurement is also ?
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Paper 3, Section II, A
2012 commentDiscuss the consequences of indistinguishability for a quantum mechanical state consisting of two identical, non-interacting particles when the particles have (a) spin zero, (b) spin 1/2.
The stationary Schrödinger equation for one particle in the potential
has normalised, spherically-symmetric real wavefunctions and energy eigenvalues with . The helium atom can be modelled by considering two non-interacting spin 1/2 particles in the above potential. What are the consequences of the Pauli exclusion principle for the ground state? Write down the two-electron state for this model in the form of a spatial wavefunction times a spin state. Assuming that wavefunctions are spherically-symmetric, find the states of the first excited energy level of the helium atom. What combined angular momentum quantum numbers does each state have?
Assuming standard perturbation theory results, arrive at a multi-dimensional integral in terms of the one-particle wavefunctions for the first-order correction to the helium ground state energy, arising from the electron-electron interaction.
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Paper 2, Section II, A
2012 comment(a) Define the Heisenberg picture of quantum mechanics in relation to the Schrödinger picture. Explain how the two pictures provide equivalent descriptions of physical results.
(b) Derive the equation of motion for an operator in the Heisenberg picture.
For a particle of mass moving in one dimension, the Hamiltonian is
where and are the position and momentum operators, and the state vector is . The eigenstates of and satisfy
Use standard methods in the Dirac formalism to show that
Calculate and express in terms of the position space wavefunction .
Write down the momentum space Hamiltonian for the potential
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Paper 1, Section II, A
2012 commentLet and be the simple harmonic oscillator annihilation and creation operators, respectively. Write down the commutator .
Consider a new operator , where with a real constant. Show that
Consider the Hamiltonian
where and are real and such that . Assuming that and , with a real constant, show that
Thus, calculate the energy of in terms of and , where is an eigenvalue of .
Assuming that , calculate in terms of and . Find the possible values of . Finally, show that the energy eigenvalues of the system are
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Paper 4, Section II, K
2012 commentFor , the pairs have independent bivariate normal distributions, with , and . The means are known; the parameters and are unknown.
Show that the joint distribution of all the variables belongs to an exponential family, and identify the natural sufficient statistic, natural parameter, and mean-value parameter. Hence or otherwise, find the maximum likelihood estimator of .
Let . What is the joint distribution of
Show that the distribution of
is . Hence describe a -level confidence interval for . Briefly explain what would change if and were also unknown.
[Recall that the distribution is that of , where, independently for and has the chi-squared distribution with degrees of freedom.]
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Paper 3, Section II,
2012 commentThe parameter vector is , with . Given , the integer random vector has a trinomial distribution, with probability mass function
Compute the score vector for the parameter , and, quoting any relevant general result, use this to determine .
Considering (1) as an exponential family with mean-value parameter , what is the corresponding natural parameter ?
Compute the information matrix for , which has -entry
where denotes the log-likelihood function, based on , expressed in terms of .
Show that the variance of is asymptotic to as . [Hint. The information matrix for is and the dispersion matrix of the maximum likelihood estimator behaves, asymptotically (for ) as .]
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Paper 2, Section II,
2012 commentCarefully defining all italicised terms, show that, if a sufficiently general method of inference respects both the Weak Sufficiency Principle and the Conditionality Principle, then it respects the Likelihood Principle.
The position of a particle at time has the Normal distribution , where is the value of an unknown parameter ; and the time, , at which the particle first reaches position has probability density function
Experimenter observes , and experimenter observes , where are fixed in advance. It turns out that . What does the Likelihood Principle say about the inferences about to be made by the two experimenters?
bases his inference about on the distribution and observed value of , while bases her inference on the distribution and observed value of . Show that these choices respect the Likelihood Principle.
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Paper 1, Section II,
2012 commentProve that, if is complete sufficient for , and is a function of , then is the minimum variance unbiased estimator of .
When the parameter takes a value , observables arise independently from the exponential distribution , having probability density function
Show that the family of distributions
with probability density function
is a conjugate family for Bayesian inference about (where is the Gamma function).
Show that the expectation of , under prior distribution (1), is , where . What is the prior variance of ? Deduce the posterior expectation and variance of , given .
Let denote the limiting form of the posterior expectation of as . Show that is the minimum variance unbiased estimator of . What is its variance?
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Paper 4, Section II, J
2012 commentState and prove Fatou's lemma. [You may use the monotone convergence theorem.]
For a measure space, define to be the vector space of integrable functions on , where functions equal almost everywhere are identified. Prove that is complete for the norm ,
[You may assume that indeed defines a norm on .] Give an example of a measure space and of a sequence that converges to almost everywhere such that .
Now let
If a sequence converges to in , does it follow that If converges to almost everywhere, does it follow that ? Justify your answers.
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Paper 3, Section II, J
2012 commentCarefully state and prove the first and second Borel-Cantelli lemmas.
Now let be a sequence of events that are pairwise independent; that is, whenever . For , let . Show that .
Using Chebyshev's inequality or otherwise, deduce that if , then almost surely. Conclude that infinitely often
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Paper 2, Section II, J
2012 commentThe Fourier transform of a Lebesgue integrable function is given by
where is Lebesgue measure on the real line. For , prove that
[You may use properties of derivatives of Fourier transforms without proof provided they are clearly stated, as well as the fact that is a probability density function.]
State and prove the almost everywhere Fourier inversion theorem for Lebesgue integrable functions on the real line. [You may use standard results from the course, such as the dominated convergence and Fubini's theorem. You may also use that where , converges to in as whenever
The probability density function of a Gamma distribution with scalar parameters is given by
Let . Is integrable?
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Paper 1, Section II, J
2012 commentCarefully state and prove Jensen's inequality for a convex function , where is an interval. Assuming that is strictly convex, give necessary and sufficient conditions for the inequality to be strict.
Let be a Borel probability measure on , and suppose has a strictly positive probability density function with respect to Lebesgue measure. Let be the family of all strictly positive probability density functions on with respect to Lebesgue measure such that . Let be a random variable with distribution . Prove that the mapping
has a unique maximiser over , attained when almost everywhere.
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Paper 4, Section II, H
2012 commentWrite an essay on the finite-dimensional representations of , including a proof of their complete reducibility, and a description of the irreducible representations and the decomposition of their tensor products.
-
Paper 3, Section II, H
2012 commentShow that every complex representation of a finite group is equivalent to a unitary representation. Let be a character of some finite group and let . Explain why there are roots of unity such that
for all integers .
For the rest of the question let be the symmetric group on some finite set. Explain why whenever is coprime to the order of .
Prove that .
State without proof a formula for when is irreducible. Is there an irreducible character of degree at least 2 with for all ? Explain your answer.
[You may assume basic facts about the symmetric group, and about algebraic integers, without proof. You may also use without proof the fact that for any th root of unity
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Paper 2, Section II, H
2012 commentSuppose that is a finite group. Define the inner product of two complex-valued class functions on . Prove that the characters of the irreducible representations of form an orthonormal basis for the space of complex-valued class functions.
Suppose that is a prime and is the field of elements. Let . List the conjugacy classes of .
Let act naturally on the set of lines in the space . Compute the corresponding permutation character and show that it is reducible. Decompose this character as a sum of two irreducible characters.
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Paper 1, Section II, 19H
2012 commentWrite down the character table of .
Suppose that is a group of order 60 containing 24 elements of order 5,20 elements of order 3 and 15 elements of order 2 . Calculate the character table of , justifying your answer.
[You may assume the formula for induction of characters, provided you state it clearly.]
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Paper 3, Section II, I
2012 commentLet be the lattice the torus , and the Weierstrass elliptic function with respect to .
(i) Let be the point given by . Determine the group
(ii) Show that defines a degree 4 holomorphic map , which is invariant under the action of , that is, for any and any . Identify a ramification point of distinct from which is fixed by every element of .
[If you use the Monodromy theorem, then you should state it correctly. You may use the fact that , and may assume without proof standard facts about .]
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Paper 2, Section II, I
2012 commentLet be the algebraic curve in defined by the polynomial where is a natural number. Using the implicit function theorem, or otherwise, show that there is a natural complex structure on . Let be the function defined by . Show that is holomorphic. Find the ramification points and the corresponding branching orders of .
Assume that extends to a holomorphic map from a compact Riemann surface to the Riemann sphere so that and that has no ramification points in . State the Riemann-Hurwitz formula and apply it to to calculate the Euler characteristic and the genus of .
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Paper 1, Section II, I
2012 comment(i) Let . Show that the unit circle is the natural boundary of the function element .
(ii) Let ; explain carefully how a holomorphic function may be defined on satisfying the equation
Let denote the connected component of the space of germs (of holomorphic functions on corresponding to the function element , with associated holomorphic . Determine the number of points of in when (a) , and (b) .
[You may assume any standard facts about analytic continuations that you may need.]
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Paper 4, Section I, K
2012 commentDefine the concepts of an exponential dispersion family and the corresponding variance function. Show that the family of Poisson distributions with parameter is an exponential dispersion family. Find the corresponding variance function and deduce from it expressions for and when . What is the canonical link function in this case?
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Paper 3, Section I, 5K
2012 commentConsider the linear model
for , where the are independent and identically distributed with distribution. What does it mean for the pair and to be orthogonal? What does it mean for all the three parameters and to be mutually orthogonal? Give necessary and sufficient conditions on so that and are mutually orthogonal. If are mutually orthogonal, find the joint distribution of the corresponding maximum likelihood estimators and .
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Paper 2, Section I, K
2012 commentThe purpose of the following study is to investigate differences among certain treatments on the lifespan of male fruit flies, after allowing for the effect of the variable 'thorax length' (thorax) which is known to be positively correlated with lifespan. Data was collected on the following variables:
longevity lifespan in days
thorax (body) length in
treat a five level factor representing the treatment groups. The levels were labelled as follows: "00", "10", "80", "11", "81".
No interactions were found between thorax length and the treatment factor. A linear model with thorax as the covariate, treat as a factor (having the above 5 levels) and longevity as the response was fitted and the following output was obtained. There were 25 males in each of the five groups, which were treated identically in the provision of fresh food.
Coefficients :
Residual standard error: on 119 degrees of freedom
Multiple R-Squared: , Adjusted R-squared:
F-statistics: on 5 and 119 degrees of freedom, p-value: 0
(a) Assuming the same treatment, how much longer would you expect a fly with a thorax length greater than another to live?
(b) What is the predicted difference in longevity between a male fly receiving treatment treat 10 and treat81 assuming they have the same thorax length?
(c) Because the flies were randomly assigned to the five groups, the distribution of thorax lengths in the five groups are essentially equal. What disadvantage would the investigators have incurred by ignoring the thorax length in their analysis (i.e., had they done a one-way ANOVA instead)?
(d) The residual-fitted plot is shown in the left panel of Figure 1 overleaf. Is it possible to determine if the regular residuals or the studentized residuals have been used to construct this plot? Explain.
(e) The Box-Cox procedure was used to determine a good transformation for this data. The plot of the log-likelihood for is shown in the right panel of Figure 1 . What transformation should be used to improve the fit and yet retain some interpretability?

Figure 1: Residual-Fitted plot on the left and Box-Cox plot on the right
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Paper 1, Section I, K
2012 commentLet be independent with , and
where is a vector of regressors and is a vector of parameters. Write down the likelihood of the data as a function of . Find the unrestricted maximum likelihood estimator of , and the form of the maximum likelihood estimator under the logistic model (1).
Show that the deviance for a comparison of the full (saturated) model to the generalised linear model with canonical link (1) using the maximum likelihood estimator can be simplified to
Finally, obtain an expression for the deviance residual in this generalised linear model.
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Paper 4, Section II, K
2012 commentLet be jointly independent and identically distributed with and conditional on .
(a) Write down the likelihood of the data , and find the maximum likelihood estimate of . [You may use properties of conditional probability/expectation without providing a proof.]
(b) Find the Fisher information for a single observation, .
(c) Determine the limiting distribution of . [You may use the result on the asymptotic distribution of maximum likelihood estimators, without providing a proof.]
(d) Give an asymptotic confidence interval for with coverage using your answers to (b) and (c).
(e) Define the observed Fisher information. Compare the confidence interval in part (d) with an asymptotic confidence interval with coverage based on the observed Fisher information.
(f) Determine the exact distribution of and find the true coverage probability for the interval in part (e). [Hint. Condition on and use the following property of conditional expectation: for random vectors, any suitable function , and ,
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Paper 1, Section II, K
2012 commentThe treatment for a patient diagnosed with cancer of the prostate depends on whether the cancer has spread to the surrounding lymph nodes. It is common to operate on the patient to obtain samples from the nodes which can then be analysed under a microscope. However it would be preferable if an accurate assessment of nodal involvement could be made without surgery. For a sample of 53 prostate cancer patients, a number of possible predictor variables were measured before surgery. The patients then had surgery to determine nodal involvement. We want to see if nodal involvement can be accurately predicted from the available variables and determine which ones are most important. The variables take the values 0 or 1 .
An indicator no yes of nodal involvement.
aged The patient's age, split into less than and 60 or over .
stage A measurement of the size and position of the tumour observed by palpation with the fingers. A serious case is coded as 1 and a less serious case as 0 .
grade Another indicator of the seriousness of the cancer which is determined by a pathology reading of a biopsy taken by needle before surgery. A value of 1 indicates a more serious case of cancer.
xray Another measure of the seriousness of the cancer taken from an X-ray reading. A value of 1 indicates a more serious case of cancer.
acid The level of acid phosphatase in the blood serum where high and low.
A binomial generalised linear model with a logit link was fitted to the data to predict nodal involvement and the following output obtained:

Part II, 2012 List of Questions
[TURN OVER (a) Give an interpretation of the coefficient of xray.
(b) Give the numerical value of the sum of the squared deviance residuals.
(c) Suppose that the predictors, stage, grade and xray are positively correlated. Describe the effect that this correlation is likely to have on our ability to determine the strength of these predictors in explaining the response.
(d) The probability of observing a value of under a Chi-squared distribution with 52 degrees of freedom is . What does this information tell us about the null model for this data? Justify your answer.
(e) What is the lowest predicted probability of the nodal involvement for any future patient?
(f) The first plot in Figure 1 shows the (Pearson) residuals and the fitted values. Explain why the points lie on two curves.
(g) The second plot in Figure 1 shows the value of where indicates that patient was dropped in computing the fit. The values for each predictor, including the intercept, are shown. Could a single case change our opinion of which predictors are important in predicting the response?

Figure 1: The plot on the left shows the Pearson residuals and the fitted values. The plot on the right shows the changes in the regression coefficients when a single point is omitted for each predictor.
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Paper 4, Section II, C
2012 commentNon-relativistic electrons of mass are confined to move in a two-dimensional plane of area . Each electron has two spin states. Compute the density of states and show that it is constant.
Write down expressions for the number of particles and the average energy of a gas of fermions in terms of the temperature and chemical potential . Find an expression for the Fermi Energy in terms of .
For , you may assume that the chemical potential does not change with temperature. Compute the low temperature heat capacity of a gas of fermions. [You may use the approximation that, for large ,
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Paper 3, Section II, C
2012 commentA ferromagnet has magnetization order parameter and is at temperature . The free energy is given by
where and are positive constants. Find the equilibrium value of the magnetization at both high and low temperatures.
Evaluate the free energy of the ground state as a function of temperature. Hence compute the entropy and heat capacity. Determine the jump in the heat capacity and identify the order of the phase transition.
After imposing a background magnetic field , the free energy becomes
Explain graphically why the system undergoes a first-order phase transition at low temperatures as changes sign.
The spinodal point occurs when the meta-stable vacuum ceases to exist. Determine the temperature of the spinodal point as a function of and .
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Paper 2, Section II, C
2012 commentExplain what is meant by an isothermal expansion and an adiabatic expansion of a gas.
By first establishing a suitable Maxwell relation, show that
and
The energy in a gas of blackbody radiation is given by , where is a constant. Derive an expression for the pressure .
Show that if the radiation expands adiabatically, is constant.
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Paper 1, Section II, C
2012 commentA meson consists of two quarks, attracted by a linear potential energy
where is the separation between the quarks and is a constant. Treating the quarks classically, compute the vibrational partition function that arises from the separation of quarks. What is the average separation of the quarks at temperature ?
Consider an ideal gas of these mesons that have the orientation of the quarks fixed so the mesons do not rotate. Compute the total partition function of the gas. What is its heat capacity ?
[Note: ]
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Paper 1, Section II, J
2012 commentConsider a multi-period binomial model with a risky asset and a riskless asset . In each period, the value of the risky asset is multiplied by if the period was good, and by otherwise. The riskless asset is worth at time , where .
(i) Assuming that and that
show how any contingent claim to be paid at time 1 can be priced and exactly replicated. Briefly explain the significance of the condition (1), and indicate how the analysis of the single-period model extends to many periods.
(ii) Now suppose that . We assume that , and that the risky asset is worth at time zero. Find the time- 0 value of an American put option with strike price and expiry at time , and find the optimal exercise policy. (Assume that the option cannot be exercised immediately at time zero.)
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Paper 4, Section II, J
2012 commentIn a one-period market, there are risky assets whose returns at time 1 are given by a column vector . The return vector has a multivariate Gaussian distribution with expectation and non-singular covariance matrix . In addition, there is a bank account giving interest , so that one unit of cash invested at time 0 in the bank account will be worth units of cash at time 1 .
An agent with the initial wealth invests in risky assets and keeps the remainder in the bank account. The return on the agent's portfolio is
The agent's utility function is , where is a parameter. His objective is to maximize .
(i) Find the agent's optimal portfolio and its expected return.
[Hint. Relate to and
(ii) Under which conditions does the optimal portfolio that you found in (i) require borrowing from the bank account?
(iii) Find the optimal portfolio if it is required that all of the agent's wealth be invested in risky assets.
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Paper 3, Section II, J
2012 comment(i) Let be a filtration. Give the definition of a martingale and a stopping time with respect to the filtration .
(ii) State Doob's optional stopping theorem. Give an example of a martingale and a stopping time such that .
(iii) Let be a standard random walk on , that is, , where are i.i.d. and or with probability .
Let where is a positive integer. Show that for all ,
Carefully justify all steps in your derivation.
[Hint. For all find such that is a martingale. You may assume that is almost surely finite.]
Let . By introducing a suitable martingale, compute .
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Paper 2, Section II, J
2012 comment(i) Give the definition of Brownian motion.
(ii) The price of an asset evolving in continuous time is represented as
where is a standard Brownian motion and and are constants. If riskless investment in a bank account returns a continuously compounded rate of interest , derive the Black-Scholes formula for the time-0 price of a European call option on asset with strike price and expiry . [Standard results from the course may be used without proof but must be stated clearly.]
(iii) In the same financial market, a certain contingent claim pays at time , where . Find the closed-form expression for the time- 0 value of this contingent claim.
Show that for every and ,
Using this identity, how would you replicate (at least approximately) the contingent claim with a portfolio consisting only of European calls?
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Paper 4, Section I,
2012 commentLet be real numbers and suppose that are distinct. Suppose that the formula
is valid for every polynomial of degree . Prove the following:
(i) for each .
(ii) .
(iii) are the roots of the Legendre polynomial of degree .
[You may assume standard orthogonality properties of the Legendre polynomials.]
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Paper 3, Section I,
2012 commentState and prove Liouville's theorem concerning approximation of algebraic numbers by rationals.
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Paper 2, Section I,
2012 comment(a) Let be a continuous map such that . Define the winding number of about the origin. State precisely a theorem about homotopy invariance of the winding number.
(b) Let be a continuous map such that is bounded as . Prove that there exists a complex number such that
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Paper 1, Section I, F
2012 commentState a version of the Baire category theorem for a complete metric space. Let be the set of real numbers with the property that, for each positive integer , there exist integers and with such that
Is an open subset of ? Is a dense subset of ? Justify your answers.
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Paper 2, Section II, F
2012 comment(a) State Runge's theorem about uniform approximability of analytic functions by complex polynomials.
(b) Let be a compact subset of the complex plane.
(i) Let be an unbounded, connected subset of . Prove that for each , the function is uniformly approximable on by a sequence of complex polynomials.
[You may not use Runge's theorem without proof.]
(ii) Let be a bounded, connected component of . Prove that there is no point such that the function is uniformly approximable on by a sequence of complex polynomials.
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Paper 3, Section II, F
2012 commentState Brouwer's fixed point theorem on the plane, and also an equivalent version of it concerning continuous retractions. Prove the equivalence of the two statements.
Let be a continuous map with the property that whenever . Show that has a fixed point. [Hint. Compose with the map that sends to the nearest point to inside the closed unit disc.]
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Paper 4, Section II, 38D
2012 commentThe shallow-water equations
describe one-dimensional flow in a channel with depth and velocity , where is the acceleration due to gravity.
(i) Find the speed of linearized waves on fluid at rest and of uniform depth.
(ii) Show that the Riemann invariants are constant on characteristic curves of slope in the -plane.
(iii) Use the shallow-water equations to derive the equation of momentum conservation
and identify the horizontal momentum flux .
(iv) A hydraulic jump propagates at constant speed along a straight constant-width channel. Ahead of the jump the fluid is at rest with uniform depth . Behind the jump the fluid has uniform depth , with . Determine both the speed of the jump and the fluid velocity behind the jump.
Express and as functions of . Hence sketch the pattern of characteristics in the frame of reference of the jump.
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Paper 2, Section II, 38D
2012 commentDerive the ray-tracing equations
for wave propagation through a slowly-varying medium with local dispersion relation . The meaning of the notation should be carefully explained.
A non-dispersive slowly varying medium has a local wave speed that depends only on the coordinate. State and prove Snell's Law relating the angle between a ray and the -axis to .
Consider the case of a medium with wavespeed , where and are positive constants. Find the equation of the ray that passes through the origin with wavevector , and show that it remains in the region . Sketch several rays passing through the origin.
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Paper 3, Section II, 39D
2012 commentThe function satisfies the equation
where is a constant. Find the dispersion relation for waves of frequency and wavenumber . Sketch a graph showing both the phase velocity and the group velocity , and state whether wave crests move faster or slower than a wave packet.
Suppose that is real and given by a Fourier transform as
Use the method of stationary phase to obtain an approximation for for fixed and large . If, in addition, , deduce an approximation for the sequence of times at which .
What can be said about if ? [Detailed calculation is not required in this case.]
[You may assume that for ]
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Paper 1, Section II, 39D
2012 commentWrite down the linearized equations governing motion in an inviscid compressible fluid and, assuming an adiabatic relationship , derive the wave equation for the velocity potential . Obtain from these linearized equations the energy equation
and give expressions for the acoustic energy density and the acoustic intensity, or energyflux vector, I.
An inviscid compressible fluid occupies the half-space , and is bounded by a very thin flexible membrane of negligible mass at an undisturbed position . Small acoustic disturbances with velocity potential in the fluid cause the membrane to be deflected to . The membrane is supported by springs that, in the deflected state, exert a restoring force on an element of the membrane. Show that the dispersion relation for waves proportional to propagating freely along the membrane is
where is the density of the fluid and is the sound speed. Show that in such a wave the component of mean acoustic intensity perpendicular to the membrane is zero.
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Paper 3, Section II, H
2012 commentSuppose that is a single observation drawn from the uniform distribution on the interval , where is unknown and might be any real number. Given we wish to test against . Let be the test which accepts if and only if , where
Show that this test has size .
Now consider
Prove that both and specify confidence intervals for . Find the confidence interval specified by when .
Let be the length of the confidence interval specified by . Let be the probability of the Type II error of . Show that
Here is an indicator variable for event . The expectation is over . [Orders of integration and expectation can be interchanged.]
Use what you know about constructing best tests to explain which of the two confidence intervals has the smaller expected length when .
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Paper 1, Section I, B
2012 commentState how to find the stationary points of a function restricted to the circle , using the method of Lagrange multipliers. Explain why, in general, the method of Lagrange multipliers works, in the case where there is just one constraint.
Find the stationary points of restricted to the circle .
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Paper 3, Section I, B
2012 commentFor a particle of unit mass moving freely on a unit sphere, the Lagrangian in polar coordinates is
Find the equations of motion. Show that is a conserved quantity, and use this result to simplify the equation of motion for . Deduce that
is a conserved quantity. What is the interpretation of ?
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Paper 2, Section II, B
2012 comment(i) A two-dimensional oscillator has action
Find the equations of motion as the Euler-Lagrange equations associated to , and use them to show that
is conserved. Write down the general solution of the equations of motion in terms of sin and , and evaluate in terms of the coefficients which arise in the general solution.
(ii) Another kind of oscillator has action
where and are real constants. Find the equations of motion and use these to show that in general is not conserved. Find the special value of the ratio for which is conserved. Explain what is special about the action in this case, and state the interpretation of .
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Paper 4, Section II, B
2012 commentConsider a functional
where is smooth in all its arguments, is a function and . Consider the function where is a small function which vanishes at and . Obtain formulae for the first and second variations of about the function . Derive the Euler-Lagrange equation from the first variation, and state its variational interpretation.
Suppose now that
where and . Find the Euler-Lagrange equation and the formula for the second variation of . Show that the function makes stationary, and that it is a (local) minimizer if .
Show that when , the function is not a minimizer of .